Authors’ Note
This paper was written by the authors with the assistance of large language models, which were used to suggest mathematical arguments, draft and revise exposition, and write code for computational verification. We have taken care to supervise this writing to ensure that the proofs meet a minimum standard of readability and that previous literature is appropriately cited. However, the resulting document remains lacking in exposition and overall coherence. We are nonetheless releasing this manuscript on Hexagon to disseminate the result as quickly as possible, because we believe it may be of interest to the statistics community. We welcome suggestions concerning mathematical corrections, omitted citations, and attribution of ideas.
P.L. was partially supported by NSF grant DMS-2450004.
Introduction
Background
Missing-data means, average treatment effects, and expected conditional covariances under random design are averages over the covariate distribution of functions of regressions given the covariates. Their estimation rates depend on what is known about both the regressions and the covariate distribution. In missing-response problems, Robins and Ritov [50 ] showed that attainable rates depend on what is known about the nuisance functions. We study rough design , in which the covariate density is unknown and subject only to fixed upper and lower bounds, while the regressions are assumed smooth. We show that below the root-n n n 4 s / ( 4 s + d ) 4s/(4s+d) 4 s / ( 4 s + d ) 2 s / d 2s/d 2 s / d s s s d d d s = d / 4 s=d/4 s = d /4
For a missing-data mean, let α \alpha α β \beta β s = ( α + β ) / 2 s=(\alpha+\beta)/2 s = ( α + β ) /2 g g g g g g γ > 0 \gamma>0 γ > 0 γ / ( 2 γ + d ) > { max ( α , β ) / d } ( d − 4 s ) / ( d + 4 s ) \gamma/(2\gamma+d)>\{\max(\alpha,\beta)/d\}(d-4s)/(d+4s) γ / ( 2 γ + d ) > { max ( α , β ) / d } ( d − 4 s ) / ( d + 4 s )
n − 1 / 2 ∨ n − 4 s / ( 4 s + d ) , (1) n^{-1/2}\vee n^{-4s/(4s+d)}, \tag*{(1)} n − 1/2 ∨ n − 4 s / ( 4 s + d ) , (1) where x ∨ y = max ( x , y ) x\vee y=\max(x,y) x ∨ y = max ( x , y ) [52 Section 3.1] , the corrected version of [53 ] , which treats the missing-data mean there and indicates the extension to other doubly robust functionals in [52 Section 11.3] . Below the threshold, the rate (1.1) cannot be improved even when g g g [49 ] ; see [52 Section 3.1] . Rates of this form are classical for quadratic functionals of a density [5 , 6 ] . For the average product of two conditional means, McGrath and Mukherjee [44 Corollaries 1 and 16] obtain this rate for a uniform or sufficiently smooth known density, and below s = d / 4 s=d/4 s = d /4
The rough-design question arises in the HOIF literature [36 , 48 , 49 , 52 ] . Immediately after their equation (1.3), Robins et al. [48, p. 338] conjecture that the rate given there is minimax, up to logarithmic factors, when the nuisance functions are equally smooth and the density smoothness γ \gamma γ γ → 0 \gamma\to0 γ → 0 n − 2 s / d log n n^{-2s/d}\log n n − 2 s / d log n n − 2 s / d n^{-2s/d} n − 2 s / d [9 , 71 ] . This limit suggests that the exponent under rough design is 2 s / d 2s/d 2 s / d
Robins et al. [52, Section 3.1] also state that heuristic arguments indicate a rate slower than (1.1) for smaller γ \gamma γ n − 2 s / d n^{-2s/d} n − 2 s / d [37 ] , that attains n − 2 s / d n^{-2s/d} n − 2 s / d n − 2 s / d n^{-2s/d} n − 2 s / d s < d / 4 s < d/4 s < d /4
Main results
Our main result, Theorem 2.3, gives minimax bounds in a nonparametric model in which the weight g g g 0 < g − ≤ g ≤ g + < ∞ 0 < g_{-} \le g \le g_{+} < \infty 0 < g − ≤ g ≤ g + < ∞ g − < g + g_{-} < g_{+} g − < g + [48 Section 3.1] that includes missing-data means and expected conditional covariances (Section 2.1). Under interior and nondegeneracy conditions stated in Section 2.2, the minimax root-mean-square error r n r_n r n
log r n = { − 2 s d log n − 2 d 2 s ( d − 4 s ) log n log g + + g − g + − g − + O ( log log n ) , s < d / 4 , − 1 2 log n + O ( 1 ) , s ≥ d / 4. (2) \log r_n =
\begin{cases}
-\frac{2s}{d}\log n-\frac{2}{d}\sqrt{2s(d-4s)\log n\log\frac{\sqrt{g_{+}}+\sqrt{g_{-}}}{\sqrt{g_{+}}-\sqrt{g_{-}}}}+O(\log\log n), & s<d/4,\\
-\frac{1}{2}\log n+O(1), & s\ge d/4.
\end{cases} \tag*{(2)} log r n = { − d 2 s log n − d 2 2 s ( d − 4 s ) log n log g + − g − g + + g − + O ( log log n ) , − 2 1 log n + O ( 1 ) , s < d /4 , s ≥ d /4. (2) Below s = d / 4 s=d/4 s = d /4 e − κ log n e^{-\kappa\sqrt{\log n}} e − κ l o g n κ \kappa κ ( log n ) 1 / 4 (\log n)^{1/4} ( log n ) 1/4 w , a , b w,a,b w , a , b § 3.2 ). In particular, they apply directly to models with additional overlap or range restrictions. They also persist if the density must be C ∞ C^{\infty} C ∞ Table 3.1 .
Table 3.1. The applications in Corollary 3.2. Here β ∗ = min ( β 0 , β 1 ) \beta_*=\min(\beta_0,\beta_1) β ∗ = min ( β 0 , β 1 ) β 0 \beta_0 β 0 β 1 \beta_1 β 1
By Theorem 2.3 , the exponent 2 s / d 2s/d 2 s / d Theorem 2.3 , whether or not the two smoothness indices are equal. For equal indices, 2 s / d 2s/d 2 s / d γ → 0 \gamma\to0 γ → 0 n − 2 s / d n^{-2s/d} n − 2 s / d e − κ log n + O ( log log n ) e^{-\kappa\sqrt{\log n}+O(\log\log n)} e − κ l o g n + O ( l o g l o g n ) 1 / log n 1/\log n 1/ log n n − 2 s / d log n n^{-2s/d}\log n n − 2 s / d log n [46 ] has obtained closely related bounds independently and concurrently, with the same threshold, polynomial exponent, and constant κ \kappa κ § 2.4 .
Ideas of the proofs
We pair moment-matching priors with polynomial approximation, as in minimax estimation of nonsmooth functionals [8 , 23 , 33 , 73 , 74 ] . For an interval I = [ x − , x + ] I=[x_{-},x_{+}] I = [ x − , x + ] 0 < x − < x + 0<x_{-}<x_{+} 0 < x − < x + c I = ( x + + x − ) / 2 c_I=(x_{+}+x_{-})/2 c I = ( x + + x − ) /2 d I = ( x + − x − ) / 2 d_I=(x_{+}-x_{-})/2 d I = ( x + − x − ) /2 t t t
1 c I + d I cos t = 1 x − x + { 1 + 2 ∑ k ≥ 1 ( − ρ I ) k cos ( k t ) } , ρ I = x + − x − x + + x − (3) \frac{1}{c_I+d_I\cos t}=\frac{1}{\sqrt{x_{-}x_{+}}}\left\{1+2\sum_{k\geq1}(-\rho_I)^k\cos(kt)\right\},\qquad\rho_I=\frac{\sqrt{x_{+}}-\sqrt{x_{-}}}{\sqrt{x_{+}}+\sqrt{x_{-}}} \tag*{(3)} c I + d I cos t 1 = x − x + 1 { 1 + 2 k ≥ 1 ∑ ( − ρ I ) k cos ( k t ) } , ρ I = x + + x − x + − x − (3) (Lemma 4.1 ). Since cos ( k t ) \cos(kt) cos ( k t ) k k k c I + d I cos t c_I+d_I\cos t c I + d I cos t I I I O ( ρ I m ) O(\rho_I^m) O ( ρ I m ) m m m I = [ g − , g + ] I=[g_{-},g_{+}] I = [ g − , g + ] ρ I \rho_I ρ I ρ \rho ρ
Upper bound. Recall that ψ ( P ) = E P W + λ ∫ a b g \psi(P)=\mathbb{E}_P W+\lambda\int abg ψ ( P ) = E P W + λ ∫ ab g a , b , g a,b,g a , b , g a a a b b b L 2 ( g ) L^2(g) L 2 ( g ) K K K ∫ a b g \int abg ∫ ab g ], and it is of order K − θ K^{-\theta} K − θ θ = 2 s / d \theta=2s/d θ = 2 s / d O ( K − θ ( m + 1 ) ν ρ m ) O(K^{-\theta}(m+1)^\nu\rho^m) O ( K − θ ( m + 1 ) ν ρ m ) m m m ν \nu ν Lemma 4.3 ). We estimate these polynomials from the n n n [17 , 19 ] ; Lemma 4.4 expresses the covariance of the lifts exactly through centered derivatives. We then choose the degrees across resolutions by the variance bounds of Lemma 4.5 , without an estimate of g g g [10 , 28 ] (§ 2.4 ).
Lower bound. We compare two priors by the method of two fuzzy hypotheses, and we follow the strategy of the lower bound of Aronow and Lopatto [3 ] for a constant conditional variance, with a random covariate density, a Poissonized sample, and a matching order M M M log n \sqrt{\log n} log n § 2.4 ; see also § 6 ). Here the matching comes from a random phase. On the bump supports of the two blocks of a pair, the density equals r 0 ± h n cos Θ r_0 \pm h_n \cos\Theta r 0 ± h n cos Θ Θ \Theta Θ u , v u,v u , v ( u , v ) (u,v) ( u , v ) A u , A v A_u,A_v A u , A v ± cos ( M Θ ) \pm\cos(M\Theta) ± cos ( M Θ ) M M M a a a b b b u / w u/w u / w v / w v/w v / w ( u + v ) / w (u+v)/w ( u + v ) / w U U U V V V § 3.1 coincide (§ 8.1 ). Each likelihood coefficient that differs between the priors contains an odd number of factors of each perturbation. The signs contribute cos ( M Θ ) \cos(M\Theta) cos ( M Θ ) p u pu p u p v pv p v Θ \Theta Θ j j j j j j Θ \Theta Θ j < M j<M j < M A u A v A_uA_v A u A v M M M ρ M \rho^M ρ M (3) . With block-constant phases, the case J = 0 J=0 J = 0 r n ≥ c a n r_n \ge c a_n r n ≥ c a n a n a_n a n (12) . In the lattice refinement, we let the phase vary across the R R R M ≤ j < M + M M \le j < M+\sqrt{M} M ≤ j < M + M R 1 − M R^{1-M} R 1 − M j j j m ≍ M m \asymp M m ≍ M m 2 ≍ log n m^2 \asymp\log n m 2 ≍ log n
The common balance. Let 0 < θ < 1 / 2 0<\theta<1/2 0 < θ < 1/2 L = log n L=\log n L = log n Δ = 1 − 2 θ \Delta=1-2\theta Δ = 1 − 2 θ τ \tau τ κ \kappa κ ν \nu ν x x x log L \log L log L M M M Δ L / x \Delta L/x Δ L / x n − 2 θ n^{-2\theta} n − 2 θ e O ( L ) e^{O(\sqrt{L})} e O ( L ) j ≥ M j\ge M j ≥ M n − θ n^{-\theta} n − θ e − θ x e^{-\theta x} e − θ x ρ Δ L / x \rho^{\Delta L/x} ρ Δ L / x x ∗ = τ Δ L / θ x_*=\sqrt{\tau\Delta L/\theta} x ∗ = τ Δ L / θ
θ x + τ Δ L x = κ L + θ ( x − x ∗ ) 2 x . \theta x+\frac{\tau\Delta L}{x}=\kappa\sqrt{L}+\frac{\theta(x-x_*)^2}{x}. θ x + x τ Δ L = κ L + x θ ( x − x ∗ ) 2 . In the lower bound, M M M M ∗ = Δ L / x ∗ M_*=\Delta L/x_* M ∗ = Δ L / x ∗ θ Δ L / M + τ M − κ L = τ ( M − M ∗ ) 2 / M \theta\Delta L/M+\tau M-\kappa\sqrt{L}=\tau(M-M_*)^2/M θ Δ L / M + τ M − κ L = τ ( M − M ∗ ) 2 / M e O ( 1 / M ) e^{O(1/M)} e O ( 1/ M ) log n \log n log n [3 ] . In the upper bound, the positive shifted values of x x x O ( L ) O(\sqrt{L}) O ( L ) log 2 \log2 log 2 O ( L 1 / 4 ) O(L^{1/4}) O ( L 1/4 ) Lemma 4.6 ). In both bounds, the shift of x x x L θ / 2 L^{\theta/2} L θ /2 L L L ( m + 1 ) ν (m+1)^\nu ( m + 1 ) ν m = O ( L ) m=O(\sqrt{L}) m = O ( L ) L ν / 2 L^{\nu/2} L ν /2 a n L a_nL a n L a n L ν / 2 + 1 / 4 a_nL^{\nu/2+1/4} a n L ν /2 + 1/4 L 1 / 4 L^{1/4} L 1/4 ν = 2 \nu=2 ν = 2
Related work
Higher-order influence functions and rough design. The HOIF estimators of Robins et al. [48 ] are built from projection kernels, which they write through inverse Gram matrices [48 p. 357] , and the truncation bias of the projected functional is the integral of the product of two projection residuals [48 Theorems 3.7–3.8] . When g g g g g g [48 Theorems 3.14–3.15] . Robins et al. [48, p. 381] conjecture that, for s < d / 4 s<d/4 s < d /4 n − 2 s / d n^{-2s/d} n − 2 s / d log n \log n log n g g g 1 / r ( n ) 1/r(n) 1/ r ( n ) log n / log r ( n ) \log n/\log r(n) log n / log r ( n ) n n n n − 4 s / ( 4 s + d ) n^{-4s/(4s+d)} n − 4 s / ( 4 s + d ) [48 Remark 4.1] . In their Remark 4.2, they refer to n − 2 s / d n^{-2s/d} n − 2 s / d s < d / 4 s<d/4 s < d /4 [48 Remark 4.2] . This effect is the overlap-weighted effect of Table 3.1 , and we compare our result for it with their conjecture in Appendix D.§ 3 .
Our estimator has a form similar to the estimators in the conjecture on p. 381 of [48 ] : it is a sum of polynomials in cell moments, estimated without bias by U-statistics, with orders up to a multiple of log n \log n log n g g g [ g − , g + ] [g_-,g_+] [ g − , g + ] § 5.1 ). Further, its order decreases as the resolution increases. Keeping high orders only at coarse resolutions resembles the truncation of the higher-order kernels in [52 Section 9, (9.6)] (§ 5.4 ). Since our estimator has root-mean-square error o ( n − 2 s / d ) o(n^{-2s/d}) o ( n − 2 s / d ) [37 ] avoid estimating g g g k = o ( n ) k=o(n) k = o ( n )
Polynomial approximation of a bilinear form in the inverse of a Gram matrix, with each term estimated by a U-statistic, was used by Kong and Valiant [28 Section 2.2] . In high-dimensional linear regression with unknown covariance Σ \Sigma Σ E [ Y X T ] Σ − 1 E [ X Y ] \mathbb{E}[YX^{\mathsf{T}}]\Sigma^{-1}\mathbb{E}[XY] E [ Y X T ] Σ − 1 E [ X Y ] E [ Y X T ] Σ j E [ X Y ] \mathbb{E}[YX^{\mathsf{T}}]\Sigma^{j}\mathbb{E}[XY] E [ Y X T ] Σ j E [ X Y ] [10 Section 3.2] use Chebyshev polynomials in Σ \Sigma Σ g g g [ g − , g + ] [g_{-},g_{+}] [ g − , g + ] Theorem 2.3 .
Estimators that do not use density smoothness. In dimension one, the difference-based estimator of Robins et al. [48 p. 380] , which follows Wang et al. [71 ] , has bias O ( n − 2 s ) O(n^{-2s}) O ( n − 2 s ) O ( n − 1 ) O(n^{-1}) O ( n − 1 ) n − 1 / 2 ∨ n − 2 s n^{-1/2}\vee n^{-2s} n − 1/2 ∨ n − 2 s min { 2 s , 1 / 2 } \min\{2s,1/2\} min { 2 s , 1/2 } Theorem 2.3 when d = 1 d=1 d = 1 n n n s = 1 / 4 s=1/4 s = 1/4 [43 Section 4.2 and Theorem 1] construct an estimator of the expected conditional covariance whose rate in probability is n − 2 s / d n^{-2s/d} n − 2 s / d s < d / 4 s<d/4 s < d /4 Corollary 3.2 on P b i l ( α , β ) \mathcal{P}_{\mathrm{bil}}(\alpha,\beta) P bil ( α , β ) § 4.2 ). Unlike the estimator of McClean et al., ours uses α , β \alpha,\beta α , β g − , g + g_{-},g_{+} g − , g +
Above the threshold, root-n n n [45 ] and empirical HOIF estimators [37 ] attain semiparametric efficiency under their respective approximation and nuisance conditions, and the bounds of [48 p. 380] recalled above give root-n n n s = d / 4 s=d/4 s = d /4 [43 Section 4.2 and Theorem 1] obtain root-n n n s > d / 4 s>d/4 s > d /4 n n n s ≥ d / 4 s\ge d/4 s ≥ d /4 Theorem 2.3 . In his setting, Park [46 Theorem 1] also obtains root-n n n s ≥ d / 4 s\ge d/4 s ≥ d /4 s = d / 4 s=d/4 s = d /4 [41 ] , and higher-order estimators of treatment parameters defined implicitly by estimating equations in [76 ] .
Structure-agnostic models. In the structure-agnostic models of Balakrishnan et al. [4 ] , in which only convergence rates are imposed on black-box nuisance estimators, first-order estimators are minimax optimal for the expected conditional covariance and, when the squared pilot-error radius is at least of order n − 1 n^{-1} n − 1 [24 ] prove the corresponding optimality for the average treatment effect, weighted average treatment effects, and the average treatment effect on the treated. Their subsequent generalization treats linear functionals of generalized regressions and includes a structure-agnostic lower bound for the expected conditional covariance even under a partially linear outcome restriction [25 Theorems 6.1–6.2 and 7.5] . Bonvini et al. [7 ] study classes that add smoothness constraints to structure-agnostic ones. The Hölder smoothness that we assume is what allows higher-order estimators to improve on first-order ones in the minimax rate. Minimax-rate optimality does not imply admissibility: in the structure-agnostic model, Liu, Mukherjee, and Robins [40, Theorems 0–2] show, under their other stated conditions and when the squared pilot-error radius is at least a constant multiple of n − 1 / 2 n^{-1/2} n − 1/2
Moment matching and polynomial approximation. Recall that, as in minimax estimation of nonsmooth functionals [8 , 23 , 33 , 73 , 74 ] , we pair moment-matching priors with polynomial approximation, with the reciprocal in place of the nonsmooth function. To estimate the support size of a discrete distribution, Wu and Yang [74 ] use best polynomial approximation of the reciprocal on an interval bounded away from zero. Our priors use the framework of the lower bounds of Robins et al. [49, Theorem 2.1 and Sections 3–4] for these functionals: independent, randomly signed smooth bumps on the cells of a partition, with cell masses that do not depend on the prior parameters, and finite response spaces. Our lower bound follows the strategy of the lower bound of Aronow and Lopatto [3 ] for a constant conditional variance under rough design. There, the covariate density is random under both priors, approximate moment matching makes the local laws of the data under the two priors close simultaneously for every number k ≤ M k \le M k ≤ M M ≍ log n M \asymp\sqrt{\log n} M ≍ log n e − C log n e^{-C\sqrt{\log n}} e − C l o g n C C C [3 ] , what is new here is the random phase of the density, under which the two priors differ only at frequency M M M M M M κ \kappa κ (2) , which the lower bound of [46 ] also identifies for the expected conditional covariance, and the lattice phases of Section 6 gain a further factor log n \log n log n
Variance estimation. In the semiparametric model Var ( Y ∣ X ) = σ 2 \operatorname{Var}(Y \mid X)=\sigma^2 Var ( Y ∣ X ) = σ 2 [9 , 71 ] when 0 < s ≤ 1 0 < s \le1 0 < s ≤ 1 s < d / 4 s < d/4 s < d /4 [48 Section 4] for s < min { 1 , d / 4 } s < \min\{1,d/4\} s < min { 1 , d /4 } [3 , 59 ] . For s > 1 s > 1 s > 1 d > 4 s d > 4s d > 4 s [3 ] identify the minimax exponent 2 ( s + 1 ) / ( d + 4 ) 2(s+1)/(d+4) 2 ( s + 1 ) / ( d + 4 ) [14 ] give a two-scale estimator that attains it and whose analysis uses only upper and lower bounds on the design density. We discuss these results in § 4.2 ; they do not apply to the nonparametric models of this paper.
Discrete covariates. In a discrete analogue, Zeng et al. [75 ] give nearly matching minimax bounds for the average treatment effect with high-dimensional discrete covariates. Their lower bound [75 Section 3.2] uses priors on the covariate probabilities, propensities, and regressions of the categories that match moments up to order log n \log n log n [75 Section 5] .
Concurrent work. Park [46 ] studies the expected conditional covariance E [ Cov ( A , Y ∣ X ) ] \mathbb{E}[\operatorname{Cov}(A,Y \mid X)] E [ Cov ( A , Y ∣ X )] [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d 0 < g − < 1 < g + 0 < g_{-} < 1 < g_{+} 0 < g − < 1 < g + E [ A ∣ X ] \mathbb{E}[A \mid X] E [ A ∣ X ] E [ Y ∣ X ] \mathbb{E}[Y \mid X] E [ Y ∣ X ] α \alpha α β \beta β U = A U=A U = A V = Y V=Y V = Y P b i l ( α , β ) \mathcal{P}_{\mathrm{bil}}(\alpha,\beta) P bil ( α , β ) § 4.2 , but the Hölder norms differ: Park requires extensions to a fixed open neighborhood of [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d [46 ] , submitted to arXiv on 4 October 2026, were developed independently and concurrently. Park states his bounds for the mean squared error. His exponent ( α + β ) / d (\alpha+\beta)/d ( α + β ) / d log { ( g + + g − ) / ( g + − g − ) } \log\{(\sqrt{g_{+}}+\sqrt{g_{-}})/(\sqrt{g_{+}}-\sqrt{g_{-}})\} log {( g + + g − ) / ( g + − g − )} θ \theta θ τ \tau τ ⌈ t ⌉ − 1 \lceil t\rceil-1 ⌈ t ⌉ − 1 d d d q t q_t q t 0 < θ < 1 / 2 0<\theta<1/2 0 < θ < 1/2
n − 2 θ ( log n ) θ e − 4 τ θ ( 1 − 2 θ ) log n and n − 2 θ ( log n ) θ + q α + q β + 1 / 2 e − 4 τ θ ( 1 − 2 θ ) log n , n^{-2\theta}(\log n)^{\theta}e^{-4\sqrt{\tau\theta(1-2\theta)\log n}}
\quad\text{and}\quad
n^{-2\theta}(\log n)^{\theta+q_{\alpha}+q_{\beta}+1/2}e^{-4\sqrt{\tau\theta(1-2\theta)\log n}}, n − 2 θ ( log n ) θ e − 4 τ θ ( 1 − 2 θ ) l o g n and n − 2 θ ( log n ) θ + q α + q β + 1/2 e − 4 τ θ ( 1 − 2 θ ) l o g n , and he obtains the order n − 1 n^{-1} n − 1 θ ≥ 1 / 2 \theta\geq1/2 θ ≥ 1/2 θ = 1 / 2 \theta=1/2 θ = 1/2 [48 Remark 4.2] . That remark concerns the variance-weighted average treatment effect, a ratio whose numerator is the expected conditional covariance [48 Example 1c] ; the conjecture after their equation (1.3), which we recall in § 2.1 , is a different passage. Since 2 τ θ ( 1 − 2 θ ) 2\sqrt{\tau\theta(1-2\theta)} 2 τ θ ( 1 − 2 θ ) κ \kappa κ (12) , the threshold, the polynomial exponent, and the constant in the stretched-exponential factor agree with those of Theorem 2.3 . In root-mean-square error, Park’s upper bound is C a n ( log n ) ν / 2 + 1 / 4 C\,a_n(\log n)^{\nu/2+1/4} C a n ( log n ) ν /2 + 1/4 c a n c\,a_n c a n c a n log n c\,a_n\log n c a n log n log n \log n log n P b i l ( α , β ) \mathcal{P}_{\mathrm{bil}}(\alpha,\beta) P bil ( α , β ) § 12 ) have Rademacher responses with conditional means u u u v v v C ∞ C^\infty C ∞ [ g − , g + ] [g_-,g_+] [ g − , g + ] u , v u,v u , v C ∞ C^\infty C ∞ [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d ϵ \epsilon ϵ c a n log n c\,a_n\log n c a n log n ( log n ) ν / 2 − 3 / 4 (\log n)^{\nu/2-3/4} ( log n ) ν /2 − 3/4 ( log n ) ν / 2 + 1 / 4 (\log n)^{\nu/2+1/4} ( log n ) ν /2 + 1/4 [46 ] , and neither work determines the power of log n \log n log n
The two proofs share their main ingredients. Both use polynomial approximation of the reciprocal on [ g − , g + ] [g_-,g_+] [ g − , g + ] 2 d 2^d 2 d [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] ∫ a b g \int abg ∫ ab g [46 ] and pairs of blocks here, and both choose the priors so that regions holding few observations do not distinguish them. The main text of [46 ] does not describe the lower-bound construction in more detail.
Park also shows that a double cross-fit local polynomial estimator whose degree depends only on d d d α \alpha α β \beta β ]. Further, his paper states the lower bound also for conditional covariances that are smooth and bounded away from zero and, when α = β \alpha=\beta α = β E [ Cov ( A , Y ∣ X ) ] / E [ Var ( Y ∣ X ) ] \mathbb{E}[\operatorname{Cov}(A,Y\mid X)]/\mathbb{E}[\operatorname{Var}(Y\mid X)] E [ Cov ( A , Y ∣ X )] / E [ Var ( Y ∣ X )] log n \log n log n ( α , β ) (\alpha,\beta) ( α , β ) { α + β < d / 2 } \{\alpha+\beta<d/2\} { α + β < d /2 } Theorem 2.3 and the applications in Table 3.1 , of which the expected conditional variance and the overlap-weighted effect have counterparts in [46 ] , refines the lower bound by the factor log n \log n log n (9) and for smooth densities (Remark 2.5). The proofs of [46 ] are in a supplement that is not part of the arXiv posting, and we could not locate the supplement online as of 7 October 2026. Our comparison is based on the main text of arXiv:2610.05006v1.
Organization
Sections 2 and 3 state the model, Theorem 2.3 , and its applications. Section 4 proves the upper bounds. § 6 gives the two-point lemma for the root-n n n § 8.1 . § 8.2 completes the proof. § 9 discusses open problems, and the appendices prove the applications.
Setting and main result
Functionals and examples
We observe n n n O = ( X , Z ) O=(X,Z) O = ( X , Z ) P P P X ∈ [ 0 , 1 ] d X\in[0,1]^d X ∈ [ 0 , 1 ] d p p p Z Z Z D = D ( Z ) D=D(Z) D = D ( Z ) U = U ( Z ) U=U(Z) U = U ( Z ) V = V ( Z ) V=V(Z) V = V ( Z ) W = W ( Z ) W=W(Z) W = W ( Z ) 0 ≤ D ≤ M 0 0\le D\le M_0 0 ≤ D ≤ M 0 ∣ U ∣ , ∣ V ∣ ≤ M 0 |U|,|V|\le M_0 ∣ U ∣ , ∣ V ∣ ≤ M 0 λ ≠ 0 \lambda\ne0 λ = 0 E P [ D ∣ X ] \mathbb{E}_P[D\mid X] E P [ D ∣ X ]
ψ ( P ) = E P W + λ E P [ E P [ U ∣ X ] E P [ V ∣ X ] E P [ D ∣ X ] ] . (4) \psi(P)=\mathbb{E}_P W+\lambda\mathbb{E}_P\left[\frac{\mathbb{E}_P[U\mid X]\mathbb{E}_P[V\mid X]}{\mathbb{E}_P[D\mid X]}\right]. \tag*{(4)} ψ ( P ) = E P W + λ E P [ E P [ D ∣ X ] E P [ U ∣ X ] E P [ V ∣ X ] ] . (4) Examples 2.1 and 2.2 give the choices of ( D , U , V , W , λ ) (D,U,V,W,\lambda) ( D , U , V , W , λ )
w = E P [ D ∣ X ] , a = E P [ U ∣ X ] w , b = E P [ V ∣ X ] w , g = w p . (5) w=\mathbb{E}_P[D\mid X],\qquad a=\frac{\mathbb{E}_P[U\mid X]}{w},\qquad b=\frac{\mathbb{E}_P[V\mid X]}{w},\qquad g=wp. \tag*{(5)} w = E P [ D ∣ X ] , a = w E P [ U ∣ X ] , b = w E P [ V ∣ X ] , g = w p . (5) Then ψ ( P ) = E P W + λ ∫ [ 0 , 1 ] d a b g \psi(P)=\mathbb{E}_P W+\lambda\int_{[0,1]^d}abg ψ ( P ) = E P W + λ ∫ [ 0 , 1 ] d ab g a a a b b b g g g
The functional ψ \psi ψ [48 Section 3.1] and to the mixed-bias class of Rotnitzky, Smucler, and Robins [55 ] . For bounded candidate functions a ~ , b ~ \tilde{a},\tilde{b} a ~ , b ~ E P [ W + λ { U ~ b ~ ( X ) + V a ~ ( X ) − D a ~ ( X ) b ~ ( X ) } ] \mathbb{E}_P[W+\lambda\{\tilde{U}\tilde{b}(X)+V\tilde{a}(X)-D\tilde{a}(X)\tilde{b}(X)\}] E P [ W + λ { U ~ b ~ ( X ) + V a ~ ( X ) − D a ~ ( X ) b ~ ( X )}] ψ ( P ) \psi(P) ψ ( P ) − λ ∫ ( a ~ − a ) ( b ~ − b ) g -\lambda\int(\tilde{a}-a)(\tilde{b}-b)g − λ ∫ ( a ~ − a ) ( b ~ − b ) g [51 equation (9), p. 849, and Section 8] . The score of the average treatment effect in double/debiased machine learning has the same product-bias property [12 Section 9.4, equations (9.4.4)–(9.4.5)] . The upper bound uses the same product structure for projection errors.
Example 2.1 (Missing-data means and average treatment effects). We consider a full-data distribution P ˉ \bar{P} P ˉ ( X , R , Y ) (X,R,Y) ( X , R , Y ) R R R Y ∈ [ 0 , 1 ] Y\in[0,1] Y ∈ [ 0 , 1 ] P P P O = ( X , ( R , R Y ) ) O=(X,(R,RY)) O = ( X , ( R , R Y )) ( D , U , V ) = ( R , 1 , R Y ) (D,U,V)=(R,1,RY) ( D , U , V ) = ( R , 1 , R Y ) W = 0 W=0 W = 0 λ = 1 \lambda=1 λ = 1
w = P ( R = 1 ∣ X ) , a = 1 / w , b = E P [ Y ∣ R = 1 , X ] , g = w p , ψ ( P ) = E P [ b ( X ) ] . w=P(R=1\mid X),\qquad a=1/w,\qquad b=\mathbb{E}_P[Y\mid R=1,X],\qquad g=wp,\qquad\psi(P)=\mathbb{E}_P[b(X)]. w = P ( R = 1 ∣ X ) , a = 1/ w , b = E P [ Y ∣ R = 1 , X ] , g = w p , ψ ( P ) = E P [ b ( X )] . Under the missing-at-random (MAR) condition R ⊥ Y ∣ X R \perp Y \mid X R ⊥ Y ∣ X [56 ] , this equals the full-data mean E P Y \mathbb{E}_P Y E P Y ( X , A , Y ) (X,A,Y) ( X , A , Y ) A A A b j = E P [ Y ∣ A = j , X ] b_j = \mathbb{E}_P[Y \mid A=j,X] b j = E P [ Y ∣ A = j , X ] j = 0 , 1 j=0,1 j = 0 , 1 R = 1 { A = j } R=1_{\{A=j\}} R = 1 { A = j } ψ ( P ) = E P [ b j ( X ) ] \psi(P)=\mathbb{E}_P[b_j(X)] ψ ( P ) = E P [ b j ( X )] [18 ] , the difference E P [ b 1 ( X ) − b 0 ( X ) ] \mathbb{E}_P[b_1(X)-b_0(X)] E P [ b 1 ( X ) − b 0 ( X )]
Example 2.2 (Conditional covariance and variance). We take D = 1 D=1 D = 1 ∣ U ∣ , ∣ V ∣ ≤ 1 |U|,|V|\le1 ∣ U ∣ , ∣ V ∣ ≤ 1 m U = E P [ U ∣ X ] m_U=\mathbb{E}_P[U\mid X] m U = E P [ U ∣ X ] m V = E P [ V ∣ X ] m_V=\mathbb{E}_P[V\mid X] m V = E P [ V ∣ X ] w = 1 w=1 w = 1 g = p g=p g = p W = U V W=UV W = U V λ = − 1 \lambda=-1 λ = − 1
ψ ( P ) = E P [ U V ] − E P [ m U ( X ) m V ( X ) ] = E P [ Cov P ( U , V ∣ X ) ] . \psi(P)=\mathbb{E}_P[UV]-\mathbb{E}_P[m_U(X)m_V(X)]=\mathbb{E}_P[\operatorname{Cov}_P(U,V\mid X)]. ψ ( P ) = E P [ U V ] − E P [ m U ( X ) m V ( X )] = E P [ Cov P ( U , V ∣ X )] . If we take U = V = Y U=V=Y U = V = Y f = E P [ Y ∣ X ] f=\mathbb{E}_P[Y\mid X] f = E P [ Y ∣ X ]
ψ ( P ) = E P [ Y 2 ] − E P [ f ( X ) 2 ] = E P [ Var P ( Y ∣ X ) ] . \psi(P)=\mathbb{E}_P[Y^2]-\mathbb{E}_P[f(X)^2]=\mathbb{E}_P[\operatorname{Var}_P(Y\mid X)]. ψ ( P ) = E P [ Y 2 ] − E P [ f ( X ) 2 ] = E P [ Var P ( Y ∣ X )] . This is the smallest mean squared error achievable by predicting Y Y Y X X X W = 0 W=0 W = 0 λ = 1 \lambda=1 λ = 1 ψ ( P ) = E P [ f ( X ) 2 ] \psi(P)=\mathbb{E}_P[f(X)^2] ψ ( P ) = E P [ f ( X ) 2 ] ( E P Y ) 2 (\mathbb{E}_P Y)^2 ( E P Y ) 2 Var P ( f ( X ) ) \operatorname{Var}_P(f(X)) Var P ( f ( X )) Var P ( Y ) > 0 \operatorname{Var}_P(Y)>0 Var P ( Y ) > 0 Var P ( f ( X ) ) / Var P ( Y ) \operatorname{Var}_P(f(X))/\operatorname{Var}_P(Y) Var P ( f ( X )) / Var P ( Y ) R 2 R^2 R 2 P ( A = 1 ∣ X ) = 1 / 2 P(A=1\mid X)=1/2 P ( A = 1 ∣ X ) = 1/2 Y o b s ∈ [ 0 , 1 ] Y_{\mathrm{obs}}\in[0,1] Y obs ∈ [ 0 , 1 ] Y = ( 2 A − 1 ) ( 2 Y o b s − 1 ) Y=(2A-1)(2Y_{\mathrm{obs}}-1) Y = ( 2 A − 1 ) ( 2 Y obs − 1 ) f f f Var P ( f ( X ) ) \operatorname{Var}_P(f(X)) Var P ( f ( X )) [34 ] .
The response in the trial of Example 2.2 is obtained by centering the outcome at 1/2 in the modified-outcome construction discussed by Tian et al. [63, Section 2.1], who attribute it to Signorovitch’s thesis. Its conditional mean under balanced randomization is the CATE.
Functionals of the form (2.1) also include some instrumental-variable parameters, and we obtain further treatment-effect parameters from them through ratios and differences (Corollary 3.2).
Model and nondegeneracy
For t > 0 t>0 t > 0 k = ⌈ t ⌉ − 1 k=\lceil t\rceil-1 k = ⌈ t ⌉ − 1 ϑ = t − k ∈ ( 0 , 1 ] \vartheta=t-k\in(0,1] ϑ = t − k ∈ ( 0 , 1 ] f : [ 0 , 1 ] d → R f:[0,1]^d\to\mathbb{R} f : [ 0 , 1 ] d → R k k k
∥ f ∥ C t ( [ 0 , 1 ] d ) = max ∣ j ∣ ≤ k ∣ D j f ∣ ∞ + max ∣ j ∣ = k sup x ≠ y ∈ [ 0 , 1 ] d ∣ D j f ( x ) − D j f ( y ) ∣ ∥ x − y ∥ 2 ϑ . \|f\|_{C^t([0,1]^d)}=\max_{|j|\le k}\lvert D^j f\rvert_\infty+\max_{|j|=k}\sup_{\substack{x\ne y\in[0,1]^d}}\frac{\lvert D^j f(x)-D^j f(y)\rvert}{\|x-y\|_2^\vartheta}. ∥ f ∥ C t ([ 0 , 1 ] d ) = ∣ j ∣ ≤ k max ∣ D j f ∣ ∞ + ∣ j ∣ = k max x = y ∈ [ 0 , 1 ] d sup ∥ x − y ∥ 2 ϑ ∣ D j f ( x ) − D j f ( y )∣ . Here D j D^j D j j j j ∣ j ∣ |j| ∣ j ∣ ∥ f ∥ C t ( [ 0 , 1 ] d ) = + ∞ \|f\|_{C^t([0,1]^d)}=+\infty ∥ f ∥ C t ([ 0 , 1 ] d ) = + ∞ H t ( H ) = { f : ∥ f ∥ C t ( [ 0 , 1 ] d ) ≤ H } \mathcal{H}^{t}(H)=\{f:\|f\|_{C^{t}([0,1]^{d})}\le H\} H t ( H ) = { f : ∥ f ∥ C t ([ 0 , 1 ] d ) ≤ H } t t t t − 1 t-1 t − 1
We fix d ≥ 1 d\ge1 d ≥ 1 α , β , H , δ > 0 \alpha,\beta,H,\delta>0 α , β , H , δ > 0 0 < g − < g + < ∞ 0<g_{-}<g_{+}<\infty 0 < g − < g + < ∞ P ( α , β ) \mathcal{P}(\alpha,\beta) P ( α , β )
w ≥ δ , a ∈ H α ( H ) , b ∈ H β ( H ) , g − ≤ g ≤ g + almost everywhere . (6) w\ge\delta,\qquad a\in\mathcal{H}^{\alpha}(H),\qquad b\in\mathcal{H}^{\beta}(H),\qquad g_{-}\le g\le g_{+}\quad\text{almost everywhere}. \tag*{(6)} w ≥ δ , a ∈ H α ( H ) , b ∈ H β ( H ) , g − ≤ g ≤ g + almost everywhere . (6) These are the only restrictions on the conditional response law and the covariate density; we impose no smoothness bound on g g g δ ≤ w ≤ M 0 \delta\le w\le M_{0} δ ≤ w ≤ M 0 g − / M 0 ≤ p ≤ g + / δ g_{-}/M_{0}\le p\le g_{+}/\delta g − / M 0 ≤ p ≤ g + / δ n n n
For a real-valued parameter T T T P \mathcal{P} P
r n ( T , P ) = inf T ^ sup P ∈ P { E P [ ( T ^ − T ( P ) ) 2 ] } 1 / 2 . r_{n}(T,\mathcal{P})=\inf_{\widehat{T}}\sup_{P\in\mathcal{P}}\left\{\mathbb{E}_{P}\left[(\widehat{T}-T(P))^{2}\right]\right\}^{1/2}. r n ( T , P ) = T inf P ∈ P sup { E P [ ( T − T ( P ) ) 2 ] } 1/2 . The infimum is over all measurable estimators based on the n n n P ( α , β ) \mathcal{P}(\alpha,\beta) P ( α , β ) r n = r n ( ψ , P ( α , β ) ) r_{n}=r_{n}(\psi,\mathcal{P}(\alpha,\beta)) r n = r n ( ψ , P ( α , β )) ≲ \lesssim ≲ ≍ \asymp ≍ n n n P ( α , β ) \mathcal{P}(\alpha,\beta) P ( α , β ) d , α , β , H , δ , g − , g + , M 0 d,\alpha,\beta,H,\delta,g_{-},g_{+},M_{0} d , α , β , H , δ , g − , g + , M 0 D , U , V , W D,U,V,W D , U , V , W λ \lambda λ π 0 \pi_{0} π 0 r r r
For the lower bound, we need a law at which every constraint in (2.3) holds strictly and perturbations of the conditional law of Z Z Z a a a b b b U = D U=D U = D ψ ( P ) = E P [ W + λ V ] \psi(P)=\mathbb{E}_{P}[W+\lambda V] ψ ( P ) = E P [ W + λV ] n − 1 / 2 n^{-1/2} n − 1/2 π 0 \pi_{0} π 0 Z Z Z
w 0 = E π 0 D , a 0 = E π 0 U w 0 , b 0 = E π 0 V w 0 . (7) w_{0}=\mathbb{E}_{\pi_{0}}D,\qquad a_{0}=\frac{\mathbb{E}_{\pi_{0}}U}{w_{0}},\qquad b_{0}=\frac{\mathbb{E}_{\pi_{0}}V}{w_{0}}. \tag*{(7)} w 0 = E π 0 D , a 0 = w 0 E π 0 U , b 0 = w 0 E π 0 V . (7) If we take X X X [ 0 , 1 ] d [0,1]^{d} [ 0 , 1 ] d Z ∼ π 0 Z\sim\pi_{0} Z ∼ π 0 P 0 P_{0} P 0 w = g = w 0 w=g=w_{0} w = g = w 0 a = a 0 a=a_{0} a = a 0 b = b 0 b=b_{0} b = b 0
max { δ , g − } < w 0 < g + , ∣ a 0 ∣ < H , ∣ b 0 ∣ < H , \max\{\delta,g_{-}\}<w_{0}<g_{+},\qquad|a_{0}|<H,\qquad|b_{0}|<H, max { δ , g − } < w 0 < g + , ∣ a 0 ∣ < H , ∣ b 0 ∣ < H , and either
Cov π 0 ( U − a 0 D V − b 0 D ) is positive definite \operatorname{Cov}_{\pi_{0}}\begin{pmatrix}U-a_{0}D\\ V-b_{0}D\end{pmatrix}\text{ is positive definite} Cov π 0 ( U − a 0 D V − b 0 D ) is positive definite or
U = V , α = β , Var π 0 ( U − a 0 D ) > 0. (8) U=V,\qquad\alpha=\beta,\qquad\operatorname{Var}_{\pi_{0}}(U-a_{0}D)>0. \tag*{(8)} U = V , α = β , Var π 0 ( U − a 0 D ) > 0. (8) The first line states that P 0 P_{0} P 0 U − a 0 D U-a_{0}D U − a 0 D V − b 0 D V-b_{0}D V − b 0 D π 0 \pi_0 π 0 a a a b b b U = D U = D U = D
For r > 0 r > 0 r > 0
P r ( π 0 ) = { P ∈ P ( α , β ) : ∣ w − w 0 ∣ ≤ r , ∣ a − a 0 ∣ ≤ r , ∣ b − b 0 ∣ ≤ r almost everywhere on [ 0 , 1 ] d } . (9) \mathcal{P}_r(\pi_0) = \left\{P \in\mathcal{P}(\alpha,\beta): |w-w_0| \le r,\ |a-a_0| \le r,\ |b-b_0| \le r\ \text{almost everywhere on }[0,1]^d\right\}. \tag*{(9)} P r ( π 0 ) = { P ∈ P ( α , β ) : ∣ w − w 0 ∣ ≤ r , ∣ a − a 0 ∣ ≤ r , ∣ b − b 0 ∣ ≤ r almost everywhere on [ 0 , 1 ] d } . (9) Under (2.5), this class contains P 0 P_0 P 0 w − ≤ w ≤ w + w_- \le w \le w_+ w − ≤ w ≤ w + a − ≤ a ≤ a + a_- \le a \le a_+ a − ≤ a ≤ a + b − ≤ b ≤ b + b_- \le b \le b_+ b − ≤ b ≤ b + w − < w 0 < w + w_- < w_0 < w_+ w − < w 0 < w + a − < a 0 < a + a_- < a_0 < a_+ a − < a 0 < a + b − < b 0 < b + b_- < b_0 < b_+ b − < b 0 < b + r r r w 0 − w − w_0-w_- w 0 − w − w + − w 0 w_+-w_0 w + − w 0 a 0 − a − a_0-a_- a 0 − a − a + − a 0 a_+-a_0 a + − a 0 b 0 − b − b_0-b_- b 0 − b − b + − b 0 b_+-b_0 b + − b 0 P r ( π 0 ) \mathcal{P}_r(\pi_0) P r ( π 0 ) P ( α , β ) \mathcal{P}(\alpha,\beta) P ( α , β )
Main theorem
The rates depend on the smoothness and on the bounds g − , g + g_-,g_+ g − , g +
s = α + β 2 , θ = 2 s d , ρ = g + − g − g + + g − , τ = − log ρ . (10) s = \frac{\alpha+\beta}{2}, \qquad\theta= \frac{2s}{d}, \qquad\rho= \frac{\sqrt{g_+}-\sqrt{g_-}}{\sqrt{g_+}+\sqrt{g_-}}, \qquad\tau= -\log\rho. \tag*{(10)} s = 2 α + β , θ = d 2 s , ρ = g + + g − g + − g − , τ = − log ρ . (10) The number ρ \rho ρ [ g − , g + ] [g_-,g_+] [ g − , g + ] τ \tau τ ρ = ( g + / g − − 1 ) / ( g + / g − + 1 ) \rho= (\sqrt{g_+/g_-}-1)/(\sqrt{g_+/g_-}+1) ρ = ( g + / g − − 1 ) / ( g + / g − + 1 ) g + / g − g_+/g_- g + / g − [65 Lecture 38] . The upper bound also depends on
q t = ( d + ⌈ t ⌉ − 1 d ) ( t > 0 ) , ν = q α + q β . (11) q_t = \binom{d+\lceil t\rceil-1}{d}\quad(t>0), \qquad\nu= q_\alpha+q_\beta. \tag*{(11)} q t = ( d d + ⌈ t ⌉ − 1 ) ( t > 0 ) , ν = q α + q β . (11) The dimension q t q_t q t ⌈ t ⌉ − 1 \lceil t\rceil-1 ⌈ t ⌉ − 1 d d d q t = 1 q_t=1 q t = 1 0 < t ≤ 1 0<t\le1 0 < t ≤ 1 ν = 2 \nu=2 ν = 2 0 < α , β ≤ 1 0<\alpha,\beta\le1 0 < α , β ≤ 1 0 < θ < 1 / 2 0<\theta<1/2 0 < θ < 1/2 τ > 0 \tau>0 τ > 0
κ = 2 θ ( 1 − 2 θ ) τ , a n = a n ( θ , τ ) = n − θ ( log n ) θ / 2 e − κ log n . (12) \kappa= 2\sqrt{\theta(1-2\theta)\tau}, \qquad a_n = a_n(\theta,\tau) = n^{-\theta}(\log n)^{\theta/2}e^{-\kappa\sqrt{\log n}}. \tag*{(12)} κ = 2 θ ( 1 − 2 θ ) τ , a n = a n ( θ , τ ) = n − θ ( log n ) θ /2 e − κ l o g n . (12) We call θ = 1 / 2 \theta=1/2 θ = 1/2 s = d / 4 s=d/4 s = d /4
Theorem 2.3 (Minimax bounds). Consider the parameter (2.1) on the class (2.3).
(a) There are C > 0 C > 0 C > 0 n 0 ≥ 3 n_0 \ge3 n 0 ≥ 3 d , α , β , H , δ , g − , g + , M 0 , ∣ λ ∣ d,\alpha,\beta,H,\delta,g_-,g_+,M_0,|\lambda| d , α , β , H , δ , g − , g + , M 0 , ∣ λ ∣ ∥ W ∥ ∞ \lVert W\rVert_\infty ∥ W ∥ ∞ P ( α , β ) \mathcal{P}(\alpha,\beta) P ( α , β ) n ≥ n 0 n \ge n_0 n ≥ n 0
r n ≤ C n − 1 / 2 if θ ≥ 1 / 2 , r n ≤ C a n ( log n ) ν / 2 + 1 / 4 if 0 < θ < 1 / 2. (13) r_n \le Cn^{-1/2}\quad\text{if }\theta\ge1/2,\qquad r_n \le Ca_n(\log n)^{\nu/2+1/4}\quad\text{if }0<\theta<1/2. \tag*{(13)} r n ≤ C n − 1/2 if θ ≥ 1/2 , r n ≤ C a n ( log n ) ν /2 + 1/4 if 0 < θ < 1/2. (13) (b) Suppose that a law π 0 \pi_0 π 0 Z Z Z (8) , and fix r > 0 r>0 r > 0 c > 0 c>0 c > 0 n 0 ≥ 3 n_0\ge3 n 0 ≥ 3 d , α , β , H , δ , g − , g + , M 0 d,\alpha,\beta,H,\delta,g_-,g_+,M_0 d , α , β , H , δ , g − , g + , M 0 D , U , V , W , λ , π 0 D,U,V,W,\lambda,\pi_0 D , U , V , W , λ , π 0 r r r P ′ \mathcal{P}' P ′ P r ( π 0 ) ⊆ P ′ ⊆ P ( α , β ) \mathcal{P}_r(\pi_0)\subseteq\mathcal{P}'\subseteq\mathcal{P}(\alpha,\beta) P r ( π 0 ) ⊆ P ′ ⊆ P ( α , β ) n ≥ n 0 n\ge n_0 n ≥ n 0 θ ≥ 1 / 2 \theta\ge1/2 θ ≥ 1/2 r n ( ψ , P ′ ) ≥ c n − 1 / 2 r_n(\psi,\mathcal{P}')\ge cn^{-1/2} r n ( ψ , P ′ ) ≥ c n − 1/2 0 < θ < 1 / 2 0<\theta<1/2 0 < θ < 1/2 r n ( ψ , P ′ ) ≥ c a n log n r_n(\psi,\mathcal{P}')\ge ca_n\log n r n ( ψ , P ′ ) ≥ c a n log n
inf ψ ^ sup P ∈ P ′ P P { ∣ ψ ^ − ψ ( P ) ∣ ≥ 2 c a n log n } ≥ 1 4 . (14) \inf_{\widehat{\psi}}\sup_{P\in\mathcal{P}'}\mathbb{P}_P\{|\widehat{\psi}-\psi(P)|\ge2ca_n\log n\}\ge\frac{1}{4}. \tag*{(14)} ψ inf P ∈ P ′ sup P P { ∣ ψ − ψ ( P ) ∣ ≥ 2 c a n log n } ≥ 4 1 . (14) In particular, if such a π 0 \pi_0 π 0 n ≥ n 0 n\ge n_0 n ≥ n 0 n 0 n_0 n 0
c n − 1 / 2 ≤ r n ≤ C n − 1 / 2 ( θ ≥ 1 / 2 ) , (15) cn^{-1/2}\le r_n\le Cn^{-1/2}\qquad(\theta\ge1/2), \tag*{(15)} c n − 1/2 ≤ r n ≤ C n − 1/2 ( θ ≥ 1/2 ) , (15) c a n log n ≤ r n ≤ C a n ( log n ) ν / 2 + 1 / 4 ( 0 < θ < 1 / 2 ) , (16) ca_n\log n\le r_n\le Ca_n(\log n)^{\nu/2+1/4}\qquad(0<\theta<1/2), \tag*{(16)} c a n log n ≤ r n ≤ C a n ( log n ) ν /2 + 1/4 ( 0 < θ < 1/2 ) , (16) and the same bounds hold for r n ( ψ , P ′ ) r_n(\psi,\mathcal{P}') r n ( ψ , P ′ ) P ′ \mathcal{P}' P ′
Remark 2.4 (Logarithmic form and the remaining gap). Taking logarithms in (16) , we obtain
log r n = − θ log n − 2 θ ( 1 − 2 θ ) τ log n + O ( log log n ) , (17) \log r_n=-\theta\log n-2\sqrt{\theta(1-2\theta)}\tau\log n+O(\log\log n), \tag*{(17)} log r n = − θ log n − 2 θ ( 1 − 2 θ ) τ log n + O ( log log n ) , (17) which is (2) . The priors of Lemma 6.2 with J = 0 J=0 J = 0 r n ≥ c a n r_n\ge ca_n r n ≥ c a n (17) . The lattice phases, with J ≍ log log n J\asymp\log\log n J ≍ log log n log n \log n log n r n / a n → ∞ r_n/a_n\to\infty r n / a n → ∞ a n a_n a n ( log n ) ν / 2 − 3 / 4 (\log n)^{\nu/2-3/4} ( log n ) ν /2 − 3/4 ( log n ) 1 / 4 (\log n)^{1/4} ( log n ) 1/4 0 < α , β ≤ 1 0<\alpha,\beta\le1 0 < α , β ≤ 1 θ ↑ 1 / 2 \theta\uparrow1/2 θ ↑ 1/2 (8) approaches singularity.
Remark 2.5 (Smooth and nearly uniform densities). Every density in the lower-bound constructions of this paper is C ∞ C^\infty C ∞ n n n p p p C ∞ C^\infty C ∞ D = 1 D=1 D = 1 w = 1 w=1 w = 1 g = p g=p g = p g ± = 1 ± ω g_\pm=1\pm\omega g ± = 1 ± ω δ < 1 \delta<1 δ < 1 (8) , even the bounds 1 − ω ≤ p ≤ 1 + ω 1-\omega\le p\le1+\omega 1 − ω ≤ p ≤ 1 + ω 0 < ω < 1 0<\omega<1 0 < ω < 1 2 s / d 2s/d 2 s / d p ± = 1 ± ω p_\pm=1\pm\omega p ± = 1 ± ω [ p − , p + ] [p_-,p_+] [ p − , p + ] (17) .
Applications
Corollary 3.2 and Table 3.1 collect the applications of Theorem 2.3 , with proofs in Appendices A to D.
For an interval I = [ i − , i + ] I=[i_{-},i_{+}] I = [ i − , i + ] 0 < i − < i + < ∞ 0<i_{-}<i_{+}<\infty 0 < i − < i + < ∞ ρ I \rho_I ρ I τ I \tau_I τ I ρ \rho ρ τ \tau τ (10) computed from [ g − , g + ] = I [g_{-},g_{+}]=I [ g − , g + ] = I θ , τ > 0 \theta,\tau>0 θ , τ > 0 ν ≥ 2 \nu\geq2 ν ≥ 2
a n ( θ , τ ) = a n ( θ , τ ) log n , a ˉ n ( θ , τ , ν ) = a n ( θ , τ ) ( log n ) ν / 2 + 1 / 4 (18) a_n(\theta,\tau)=a_n(\theta,\tau)\log n,\qquad\bar{a}_n(\theta,\tau,\nu)=a_n(\theta,\tau)(\log n)^{\nu/2+1/4} \tag*{(18)} a n ( θ , τ ) = a n ( θ , τ ) log n , a ˉ n ( θ , τ , ν ) = a n ( θ , τ ) ( log n ) ν /2 + 1/4 (18) if θ < 1 / 2 \theta<1/2 θ < 1/2 a n ( θ , τ ) a_n(\theta,\tau) a n ( θ , τ ) (12) , and a ˉ n ( θ , τ ) = a ˉ n ( θ , τ , ν ) = n − 1 / 2 \bar{a}_n(\theta,\tau)=\bar{a}_n(\theta,\tau,\nu)=n^{-1/2} a ˉ n ( θ , τ ) = a ˉ n ( θ , τ , ν ) = n − 1/2 θ ≥ 1 / 2 \theta\geq1/2 θ ≥ 1/2
Definition 3.1. Let T T T P \mathcal{P} P θ > 0 \theta>0 θ > 0 ν ≥ 2 \nu\geq2 ν ≥ 2 I I I ( T , P ) (T,\mathcal{P}) ( T , P ) ( θ , I ) (\theta,I) ( θ , I ) c > 0 c>0 c > 0 n 0 ≥ 3 n_0\geq3 n 0 ≥ 3 n ≥ n 0 n\geq n_0 n ≥ n 0 r n ( T , P ) ≥ c a n ( θ , τ I ) r_n(T,\mathcal{P})\geq c a_n(\theta,\tau_I) r n ( T , P ) ≥ c a n ( θ , τ I ) θ < 1 / 2 \theta<1/2 θ < 1/2
inf T ^ sup P ∈ P { P P ∣ T ^ − T ( P ) ∣ ≥ 2 c a n ( θ , τ I ) } ≥ 1 4 . (19) \inf_{\widehat{T}}\sup_{P\in\mathcal{P}}\left\{\mathbb{P}_{P}\left\lvert\widehat{T}-T(P)\right\rvert\geq2c a_n(\theta,\tau_I)\right\}\geq\frac{1}{4}. \tag*{(19)} T inf P ∈ P sup { P P T − T ( P ) ≥ 2 c a n ( θ , τ I ) } ≥ 4 1 . (19) We say that ( T , P ) (T,\mathcal{P}) ( T , P ) upper bracket ( θ , I , ν ) (\theta,I,\nu) ( θ , I , ν ) C > 0 C>0 C > 0 n 0 ≥ 3 n_0\geq3 n 0 ≥ 3 r n ( T , P ) ≤ C a ˉ n ( θ , τ I , ν ) r_n(T,\mathcal{P})\leq C\bar{a}_n(\theta,\tau_I,\nu) r n ( T , P ) ≤ C a ˉ n ( θ , τ I , ν ) n ≥ n 0 n\geq n_0 n ≥ n 0 bracket ( θ , I , ν ) (\theta,I,\nu) ( θ , I , ν )
The targets and classes in Table 3.1 are defined below. Most of these classes bound the covariate density by
0 < p − < 1 < p + < ∞ , p − ≤ p ≤ p + . (20) 0<p_{-}<1<p_{+}<\infty,\qquad p_{-}\leq p\leq p_{+}. \tag*{(20)} 0 < p − < 1 < p + < ∞ , p − ≤ p ≤ p + . (20) We write τ p = τ [ p − , p + ] \tau_p=\tau_{[p_{-},p_{+}]} τ p = τ [ p − , p + ] κ p ( θ ) = 2 θ ( 1 − 2 θ ) τ p \kappa_p(\theta)=2\sqrt{\theta(1-2\theta)}\tau_p κ p ( θ ) = 2 θ ( 1 − 2 θ ) τ p 0 < θ < 1 / 2 0<\theta<1/2 0 < θ < 1/2 I ζ = [ ( 1 − ζ ) p − , ( 1 + ζ ) p + ] I_\zeta=[(1-\zeta)p_{-},(1+\zeta)p_{+}] I ζ = [( 1 − ζ ) p − , ( 1 + ζ ) p + ] ζ ∈ ( 0 , 1 ) \zeta\in(0,1) ζ ∈ ( 0 , 1 )
Corollary 3.2 (Applications). Fix d ∈ N d\in\mathbb{N} d ∈ N Table 3.1 . Each pair ( T , P ) (T,\mathcal{P}) ( T , P ) ( 2 s / d , I , ν ) (2s/d,I,\nu) ( 2 s / d , I , ν )
(a) On M s e p \mathcal{M}^{\mathrm{sep}} M sep T s e p \mathcal{T}^{\mathrm{sep}} T sep I = [ p − , p + ] I=[p_{-},p_{+}] I = [ p − , p + ] I ζ I_\zeta I ζ I I I ζ ∈ ( 0 , 1 ) \zeta\in(0,1) ζ ∈ ( 0 , 1 ) s < d / 4 s<d/4 s < d /4
log r n ( T , P ) = − 2 s d log n − κ p ( 2 s / d ) log n + o ( log n ) . (21) \log r_n(T,\mathcal{P})=-\frac{2s}{d}\log n-\kappa_p(2s/d)\sqrt{\log n}+o(\sqrt{\log n}). \tag*{(21)} log r n ( T , P ) = − d 2 s log n − κ p ( 2 s / d ) log n + o ( log n ) . (21) (b) Let f 0 f_0 f 0 B 0 = ∥ f 0 ∥ ∞ < ∞ B_0 = \lVert f_0\rVert_\infty< \infty B 0 = ∥ f 0 ∥ ∞ < ∞ P q u a d ( s ) \mathcal{P}_{\mathrm{quad}}(s) P quad ( s ) E P [ ( f ( X ) − f 0 ( X ) ) 2 ] \mathbb{E}_{P}[(f(X)-f_0(X))^2] E P [( f ( X ) − f 0 ( X ) ) 2 ] ( 2 s / d , [ p − , p + ] , 2 q s ) (2s/d,[p_-,p_+],2q_s) ( 2 s / d , [ p − , p + ] , 2 q s ) σ ˉ 2 \bar{\sigma}^2 σ ˉ 2
The constants c c c C C C n 0 n_0 n 0 T T T P \mathcal{P} P ζ \zeta ζ f 0 f_0 f 0 B 0 B_0 B 0
In every row, r n ( T , P ) = n − min { 1 / 2 , 2 s / d } + o ( 1 ) r_n(T,\mathcal{P})=n^{-\min\{1/2,2s/d\}+o(1)} r n ( T , P ) = n − m i n { 1/2 , 2 s / d } + o ( 1 ) ν = 2 \nu=2 ν = 2 s < d / 4 s<d/4 s < d /4 n o ( 1 ) n^{o(1)} n o ( 1 ) 4 s / ( 4 s + d ) 4s/(4s+d) 4 s / ( 4 s + d ) g g g [52 Section 3.1 and Corollary 9.1] . Every density in the lower-bound constructions is C ∞ C^\infty C ∞ p p p C ∞ C^\infty C ∞
Missing-data means and treatment effects
We use the missing-data observations O = ( X , ( R , R Y ) ) O=(X,(R,RY)) O = ( X , ( R , R Y )) ( D , U , V , W , λ ) = ( R , 1 , R Y , 0 , 1 ) (D,U,V,W,\lambda)=(R,1,RY,0,1) ( D , U , V , W , λ ) = ( R , 1 , R Y , 0 , 1 ) w = P ( R = 1 ∣ X ) w=P(R=1\mid X) w = P ( R = 1 ∣ X ) a = 1 / w a=1/w a = 1/ w b = E P [ Y ∣ R = 1 , X ] b=\mathop{\mathbb{E}}_P[Y\mid R=1,X] b = E P [ Y ∣ R = 1 , X ] E P [ b ( X ) ] \mathop{\mathbb{E}}_P[b(X)] E P [ b ( X )] 0 < η < 1 / 4 0<\eta<1/4 0 < η < 1/4 H > 2 H>2 H > 2 M η ( α , β , H ) \mathcal{M}_{\eta}(\alpha,\beta,H) M η ( α , β , H ) Y ∈ [ 0 , 1 ] Y\in[0,1] Y ∈ [ 0 , 1 ]
1 / w ∈ H α ( H ) , b ∈ H β ( H ) , η ≤ w ≤ 1 − η , η ≤ w p ≤ η − 1 . 1/w\in\mathcal{H}^{\alpha}(H),\qquad b\in\mathcal{H}^{\beta}(H),\qquad\eta\leq w\leq1-\eta,\qquad\eta\leq wp\leq\eta^{-1}. 1/ w ∈ H α ( H ) , b ∈ H β ( H ) , η ≤ w ≤ 1 − η , η ≤ w p ≤ η − 1 . This is the class P ( α , β ) \mathcal{P}(\alpha,\beta) P ( α , β ) M 0 = 1 M_0=1 M 0 = 1 δ = η \delta=\eta δ = η [ g − , g + ] = [ η , η − 1 ] [g_-,g_+]=[\eta,\eta^{-1}] [ g − , g + ] = [ η , η − 1 ] ρ = ( 1 − η ) / ( 1 + η ) \rho=(1-\eta)/(1+\eta) ρ = ( 1 − η ) / ( 1 + η ) w ≤ 1 − η w\leq1-\eta w ≤ 1 − η
For treatment effects, we observe O = ( X , A , Y ) O=(X,A,Y) O = ( X , A , Y ) A ∈ { 0 , 1 } A\in\{0,1\} A ∈ { 0 , 1 } Y ∈ [ 0 , 1 ] Y\in[0,1] Y ∈ [ 0 , 1 ]
w ( x ) = P ( A = 1 ∣ X = x ) , w 1 = w , w 0 = 1 − w , w(x)=P(A=1\mid X=x),\qquad w_1=w,\qquad w_0=1-w, w ( x ) = P ( A = 1 ∣ X = x ) , w 1 = w , w 0 = 1 − w , a j = 1 / w j , g j = w j p , b j ( x ) = E P [ Y ∣ A = j , X = x ] ( j = 0 , 1 ) . a_j=1/w_j,\qquad g_j=w_jp,\qquad b_j(x)=\mathop{\mathbb{E}}_P[Y\mid A=j,X=x]\qquad(j=0,1). a j = 1/ w j , g j = w j p , b j ( x ) = E P [ Y ∣ A = j , X = x ] ( j = 0 , 1 ) . Here the subscripts index the treatment arms; w 0 w_0 w 0 a 0 a_0 a 0 T η ( α , β 0 , β 1 , H ) \mathcal{T}_{\eta}(\alpha,\beta_0,\beta_1,H) T η ( α , β 0 , β 1 , H ) O O O
η ≤ w ≤ 1 − η , a 0 , a 1 ∈ H α ( H ) , η ≤ g 0 , g 1 ≤ η − 1 , b j ∈ H β j ( H ) ( j = 0 , 1 ) . (22) \eta\leq w\leq1-\eta,\qquad a_0,a_1\in\mathcal{H}^{\alpha}(H),\qquad\eta\leq g_0,g_1\leq\eta^{-1},\qquad b_j\in\mathcal{H}^{\beta_j}(H)\quad(j=0,1). \tag*{(22)} η ≤ w ≤ 1 − η , a 0 , a 1 ∈ H α ( H ) , η ≤ g 0 , g 1 ≤ η − 1 , b j ∈ H β j ( H ) ( j = 0 , 1 ) . (22) These restrictions imply 2 η ≤ p ≤ 2 η − 1 2\eta\leq p\leq2\eta^{-1} 2 η ≤ p ≤ 2 η − 1 μ j = E P [ b j ( X ) ] \mu_j=\mathop{\mathbb{E}}_P[b_j(X)] μ j = E P [ b j ( X )]
T A T E = μ 1 − μ 0 T_{\mathrm{ATE}}=\mu_1-\mu_0 T ATE = μ 1 − μ 0 T A T T = E P [ b 1 ( X ) − b 0 ( X ) ∣ A = 1 ] , T A T U = E P [ b 1 ( X ) − b 0 ( X ) ∣ A = 0 ] . (23) T_{\mathrm{ATT}}=\mathop{\mathbb{E}}_P[b_1(X)-b_0(X)\mid A=1],\qquad T_{\mathrm{ATU}}=\mathop{\mathbb{E}}_P[b_1(X)-b_0(X)\mid A=0]. \tag*{(23)} T ATT = E P [ b 1 ( X ) − b 0 ( X ) ∣ A = 1 ] , T ATU = E P [ b 1 ( X ) − b 0 ( X ) ∣ A = 0 ] . (23) Under consistency, conditional exchangeability, and positivity [18 ] , these are the average effects in the population, among treated units, and among untreated units. Their effective smoothness indices are
s A T E = α + min ( β 0 , β 1 ) 2 , s A T T = α + β 0 2 , s A T U = α + β 1 2 . (24) s_{\mathrm{ATE}}=\frac{\alpha+\min(\beta_0,\beta_1)}{2},\qquad s_{\mathrm{ATT}}=\frac{\alpha+\beta_0}{2},\qquad s_{\mathrm{ATU}}=\frac{\alpha+\beta_1}{2}. \tag*{(24)} s ATE = 2 α + min ( β 0 , β 1 ) , s ATT = 2 α + β 0 , s ATU = 2 α + β 1 . (24) To bound the difference from below, we vary one arm regression and keep the other constant (Appendix B).
If we bound the covariate density and the propensity separately, the correction is determined by the density interval rather than by the range of w p wp w p H 0 > 1 H_0>1 H 0 > 1 0 < ϵ < 1 / 2 0<\epsilon<1/2 0 < ϵ < 1/2 M s e p \mathcal{M}^{\mathrm{sep}} M sep Y ∈ [ 0 , 1 ] Y\in[0,1] Y ∈ [ 0 , 1 ] p p p (20) , w ∈ H α ( H 0 ) w\in\mathcal{H}^{\alpha}(H_0) w ∈ H α ( H 0 ) ϵ ≤ w ≤ 1 − ϵ \epsilon\leq w\leq1-\epsilon ϵ ≤ w ≤ 1 − ϵ b ∈ H β ( H 0 ) b\in\mathcal{H}^{\beta}(H_0) b ∈ H β ( H 0 ) T s e p \mathcal{T}^{\mathrm{sep}} T sep ( X , A , Y ) (X,A,Y) ( X , A , Y ) Y ∈ [ 0 , 1 ] Y\in[0,1] Y ∈ [ 0 , 1 ] p p p (20) , w = P ( A = 1 ∣ X ) w=P(A=1\mid X) w = P ( A = 1 ∣ X ) b j ∈ H β j ( H 0 ) b_j\in\mathcal{H}^{\beta_j}(H_0) b j ∈ H β j ( H 0 ) j = 0 , 1 j=0,1 j = 0 , 1 α \alpha α w w w 1 / w 1/w 1/ w ϵ ≤ w ≤ 1 − ϵ \epsilon\le w \le1-\epsilon ϵ ≤ w ≤ 1 − ϵ p p p
For the upper bounds on M s e p \mathcal{M}^{\mathrm{sep}} M sep T s e p \mathcal{T}^{\mathrm{sep}} T sep R R R R Y RY R Y w ^ \widehat{w} w w w w w p wp w p w p / w ^ wp/\widehat{w} w p / w 1 ± ζ 1 \pm\zeta 1 ± ζ p p p § 3 ). For μ j \mu_j μ j R = 1 { A = j } R = 1_{\{A=j\}} R = 1 { A = j } w j w_j w j 1 − w 1-w 1 − w I ζ I_\zeta I ζ κ p ( 2 s / d ) \kappa_p(2s/d) κ p ( 2 s / d ) ζ → 0 \zeta\to0 ζ → 0 (21) .
Conditional covariances and variances
We take D = 1 D=1 D = 1 g = p g=p g = p H 0 > 1 H_0>1 H 0 > 1 (20) . Let P b i l ( α , β ) \mathcal{P}_{\mathrm{bil}}(\alpha,\beta) P bil ( α , β ) ( X , U , V ) (X,U,V) ( X , U , V ) (20) , ∣ U ∣ , ∣ V ∣ ≤ 1 |U|,|V|\le1 ∣ U ∣ , ∣ V ∣ ≤ 1 m U ∈ H α ( H 0 ) m_U\in\mathcal{H}^{\alpha}(H_0) m U ∈ H α ( H 0 ) m V ∈ H β ( H 0 ) m_V\in\mathcal{H}^{\beta}(H_0) m V ∈ H β ( H 0 ) m U = E P [ U ∣ X ] m_U=\mathbb{E}_P[U\mid X] m U = E P [ U ∣ X ] m V = E P [ V ∣ X ] m_V=\mathbb{E}_P[V\mid X] m V = E P [ V ∣ X ] U = V = Y U=V=Y U = V = Y P q u a d ( s ) \mathcal{P}_{\mathrm{quad}}(s) P quad ( s ) ( X , Y ) (X,Y) ( X , Y ) (20) , ∣ Y ∣ ≤ 1 |Y|\le1 ∣ Y ∣ ≤ 1 f ∈ H s ( H 0 ) f\in\mathcal{H}^{s}(H_0) f ∈ H s ( H 0 ) f = E P [ Y ∣ X ] f=\mathbb{E}_P[Y\mid X] f = E P [ Y ∣ X ]
Q 2 = E P [ f ( X ) 2 ] , σ ˉ 2 = E P [ Var P ( Y ∣ X ) ] , E ∗ = Var P ( f ( X ) ) = Q 2 − ( E P Y ) 2 . Q_2=\mathbb{E}_P[f(X)^2],\qquad\bar{\sigma}^2=\mathbb{E}_P[\operatorname{Var}_P(Y\mid X)],\qquad E_*=\operatorname{Var}_P(f(X))=Q_2-(\mathbb{E}_P Y)^2. Q 2 = E P [ f ( X ) 2 ] , σ ˉ 2 = E P [ Var P ( Y ∣ X )] , E ∗ = Var P ( f ( X )) = Q 2 − ( E P Y ) 2 . For fixed v min ∈ ( 0 , 1 ) v_{\min}\in(0,1) v m i n ∈ ( 0 , 1 ) P q u a d ( s ; v min ) \mathcal{P}_{\mathrm{quad}}(s;v_{\min}) P quad ( s ; v m i n ) P q u a d ( s ) \mathcal{P}_{\mathrm{quad}}(s) P quad ( s ) Var P ( Y ) ≥ v min \operatorname{Var}_P(Y)\ge v_{\min} Var P ( Y ) ≥ v m i n R ∗ 2 = E ∗ / Var P ( Y ) R_*^2=E_*/\operatorname{Var}_P(Y) R ∗ 2 = E ∗ / Var P ( Y ) ( X , A , Y o b s ) (X,A,Y_{\mathrm{obs}}) ( X , A , Y obs ) P t r i a l ( s ) \mathcal{P}_{\mathrm{trial}}(s) P trial ( s ) A ∈ { 0 , 1 } A\in\{0,1\} A ∈ { 0 , 1 } Y o b s ∈ [ 0 , 1 ] Y_{\mathrm{obs}}\in[0,1] Y obs ∈ [ 0 , 1 ] P ( A = 1 ∣ X ) = 1 / 2 P(A=1\mid X)=1/2 P ( A = 1 ∣ X ) = 1/2 p p p (20) , and the conditional average treatment effect (CATE) f = E P [ Y o b s ∣ A = 1 , X ] − E P [ Y o b s ∣ A = 0 , X ] f=\mathbb{E}_P[Y_{\mathrm{obs}}\mid A=1,X]-\mathbb{E}_P[Y_{\mathrm{obs}}\mid A=0,X] f = E P [ Y obs ∣ A = 1 , X ] − E P [ Y obs ∣ A = 0 , X ] H s ( H 0 ) \mathcal{H}^{s}(H_0) H s ( H 0 )
For E ∗ E_* E ∗ R ∗ 2 R_*^2 R ∗ 2 Q 2 Q_2 Q 2 Y Y Y Y 2 Y^2 Y 2 Var P ( Y ) → 1 \operatorname{Var}_P(Y)\to1 Var P ( Y ) → 1 [69 Section 1.1] . In the trial, the ATE E P [ f ( X ) ] \mathbb{E}_P[f(X)] E P [ f ( X )] Var P ( f ( X ) ) \operatorname{Var}_P(f(X)) Var P ( f ( X )) E ∗ E_* E ∗ f f f [57 ] . In Corollary 3.2 (b), the loss E P [ ( Y − f 0 ( X ) ) 2 ] \mathbb{E}_P[(Y-f_0(X))^2] E P [( Y − f 0 ( X ) ) 2 ] σ ˉ 2 \bar{\sigma}^2 σ ˉ 2
For the expected conditional covariance, the probability bound (19) on P b i l ( α , β ) \mathcal{P}_{\mathrm{bil}}(\alpha,\beta) P bil ( α , β ) [43 ] upper bound in probability is minimax optimal. Evans and Jones [15 Section 1(a) and Section 2] studied nearest-neighbor estimators of residual moments and covariance under Lipschitz regression, regularity conditions on the design density, and additive errors independent of the covariates; Williamson et al. [72 Section 2.3, Example 1, and Section 3, Theorems 1–2] studied R 2 R^2 R 2 Var ( Y ∣ X ) = σ 2 \operatorname{Var}(Y \mid X) = \sigma^2 Var ( Y ∣ X ) = σ 2 [48 , 59 ] . There, when 0 < s ≤ 1 0 < s \le1 0 < s ≤ 1 s < d / 4 s < d/4 s < d /4 [9 , 71 ] . This was sketched in [48 Section 4, pp. 382–383] for s < min { 1 , d / 4 } s < \min\{1, d/4\} s < min { 1 , d /4 } [59 ] for errors independent of X X X [3 Theorem 1.3] for a model in which the law of the error may depend on X X X s > 1 s > 1 s > 1 [3 ] resolved the uniform minimax formulation of an open question of Robins, recorded by Richardson and Rotnitzky [47 Section 6] . For s > 1 s > 1 s > 1 d > 4 s d > 4s d > 4 s [14 Section 3.3, Proposition 3.4] has root-mean-square error O ( n − 2 ( s + 1 ) / ( d + 4 ) ) O(n^{-2(s+1)/(d+4)}) O ( n − 2 ( s + 1 ) / ( d + 4 ) ) [3 Theorem 1.1] . The proof for this estimator uses only upper and lower bounds on the covariate density [14 Appendix A.5] . Corollary 3.2 concerns the fully nonparametric model, to which these faster rates do not apply.
Wald ratios and the overlap-weighted effect
We observe ( X , Z , A , Y ) (X, Z, A, Y) ( X , Z , A , Y ) Z Z Z A A A Y ∈ [ 0 , 1 ] Y \in[0,1] Y ∈ [ 0 , 1 ] w z ( x ) = P ( Z = z ∣ X = x ) w_z(x) = P(Z = z \mid X = x) w z ( x ) = P ( Z = z ∣ X = x ) m Y , z ( x ) = E P [ Y ∣ Z = z , X = x ] m_{Y,z}(x) = \mathbb{E}_P[Y \mid Z = z, X = x] m Y , z ( x ) = E P [ Y ∣ Z = z , X = x ] m A , z ( x ) = E P [ A ∣ Z = z , X = x ] m_{A,z}(x) = \mathbb{E}_P[A \mid Z = z, X = x] m A , z ( x ) = E P [ A ∣ Z = z , X = x ] Δ Y = m Y , 1 − m Y , 0 \Delta_Y = m_{Y,1} - m_{Y,0} Δ Y = m Y , 1 − m Y , 0 Δ A = m A , 1 − m A , 0 \Delta_A = m_{A,1} - m_{A,0} Δ A = m A , 1 − m A , 0 E P [ Δ A ( X ) ] > 0 \mathbb{E}_P[\Delta_A(X)] > 0 E P [ Δ A ( X )] > 0 Wald ratio is T W = E P [ Δ Y ( X ) ] / E P [ Δ A ( X ) ] T_W = \mathbb{E}_P[\Delta_Y(X)]/\mathbb{E}_P[\Delta_A(X)] T W = E P [ Δ Y ( X )] / E P [ Δ A ( X )] P ( Z = 1 ∣ X ) = 1 / 2 P(Z = 1 \mid X) = 1/2 P ( Z = 1 ∣ X ) = 1/2 A ≤ Z A \le Z A ≤ Z q = P ( A = 1 ∣ Z = 1 , X ) q = P(A = 1 \mid Z = 1, X) q = P ( A = 1 ∣ Z = 1 , X ) Δ A = q \Delta_A = q Δ A = q average conditional Wald ratio is T C W = E P [ b ( X ) ] T_{CW} = \mathbb{E}_P[b(X)] T C W = E P [ b ( X )] b = Δ Y / q b = \Delta_Y/q b = Δ Y / q α , β > 0 \alpha, \beta> 0 α , β > 0 H > 2 H > 2 H > 2 0 < η < 1 / 4 0 < \eta< 1/4 0 < η < 1/4 0 < c W ≤ 1 0 < c_W \le1 0 < c W ≤ 1 I η , c W ( α , β , H ) \mathcal{I}_{\eta,c_W}(\alpha,\beta,H) I η , c W ( α , β , H ) z = 0 , 1 z = 0,1 z = 0 , 1
η ≤ w z ≤ 1 − η , 1 / w z ∈ H α ( H ) , η ≤ w z p ≤ η − 1 , \eta\le w_z \le1-\eta,\quad1/w_z \in\mathcal{H}^{\alpha}(H),\quad\eta\le w_zp \le\eta^{-1}, η ≤ w z ≤ 1 − η , 1/ w z ∈ H α ( H ) , η ≤ w z p ≤ η − 1 , m Y , z , m A , z ∈ H β ( H ) , E P [ Δ A ( X ) ] ≥ c W , m_{Y,z},m_{A,z} \in\mathcal{H}^{\beta}(H),\quad\mathbb{E}_P[\Delta_A(X)] \ge c_W, m Y , z , m A , z ∈ H β ( H ) , E P [ Δ A ( X )] ≥ c W , and we let J η ( α , β , H ) \mathcal{J}_{\eta}(\alpha,\beta,H) J η ( α , β , H ) P ( Z = 1 ∣ X ) = 1 / 2 P(Z = 1 \mid X) = 1/2 P ( Z = 1 ∣ X ) = 1/2 A ≤ Z A \le Z A ≤ Z
1 / q ∈ H α ( H ) , b ∈ H β ( H ) , η ≤ q ≤ 1 − η , η ≤ q p ≤ η − 1 . 1/q \in\mathcal{H}^{\alpha}(H),\quad b \in\mathcal{H}^{\beta}(H),\quad\eta\le q \le1-\eta,\quad\eta\le qp \le\eta^{-1}. 1/ q ∈ H α ( H ) , b ∈ H β ( H ) , η ≤ q ≤ 1 − η , η ≤ q p ≤ η − 1 . For observations ( X , A , Y ) (X, A, Y) ( X , A , Y ) Y ∈ [ 0 , 1 ] Y \in[0,1] Y ∈ [ 0 , 1 ] w = P ( A = 1 ∣ X ) w = P(A = 1 \mid X) w = P ( A = 1 ∣ X ) m = E P [ Y ∣ X ] m = \mathbb{E}_P[Y \mid X] m = E P [ Y ∣ X ] overlap-weighted effect is
γ O W = E P [ Cov P ( A , Y ∣ X ) ] E P [ Var P ( A ∣ X ) ] = E P [ w ( 1 − w ) ( b 1 − b 0 ) ( X ) ] E P [ w ( 1 − w ) ( X ) ] , (25) \gamma_{\mathrm{OW}} = \frac{\mathbb{E}_P[\operatorname{Cov}_P(A,Y \mid X)]}{\mathbb{E}_P[\operatorname{Var}_P(A \mid X)]} = \frac{\mathbb{E}_P[w(1-w)(b_1-b_0)(X)]}{\mathbb{E}_P[w(1-w)(X)]}, \tag*{(25)} γ OW = E P [ Var P ( A ∣ X )] E P [ Cov P ( A , Y ∣ X )] = E P [ w ( 1 − w ) ( X )] E P [ w ( 1 − w ) ( b 1 − b 0 ) ( X )] , (25) with b a = E P [ Y ∣ A = a , X ] b_a = \mathbb{E}_P[Y \mid A = a, X] b a = E P [ Y ∣ A = a , X ] A A A Y Y Y γ A + h ( X ) \gamma A + h(X) γ A + h ( X ) [54 ] for the partially linear model, which we do not assume to hold. We let P O W ( α , β ) \mathcal{P}_{\mathrm{OW}}(\alpha,\beta) P OW ( α , β ) w ∈ H α ( H 0 ) w \in\mathcal{H}^{\alpha}(H_0) w ∈ H α ( H 0 ) m ∈ H β ( H 0 ) m \in\mathcal{H}^{\beta}(H_0) m ∈ H β ( H 0 ) ϵ ≤ w ≤ 1 − ϵ \epsilon\leq w \leq1-\epsilon ϵ ≤ w ≤ 1 − ϵ H 0 > 1 H_0 > 1 H 0 > 1 0 < ϵ < 1 / 2 0 < \epsilon< 1/2 0 < ϵ < 1/2 γ O W \gamma_{\mathrm{OW}} γ OW ( α + β ) / 2 (\alpha+\beta)/2 ( α + β ) /2 α \alpha α s O W = min { ( α + β ) / 2 , α } s_{\mathrm{OW}}=\min\{(\alpha+\beta)/2,\alpha\} s OW = min {( α + β ) /2 , α } Table 3.1 .
Upper bounds
We prove Theorem 2.3 (a) by approximating ∫ a b g \int abg ∫ ab g g g g [48 , 52 ] , which correct an initial estimate by U-statistics of increasing order. Our estimator uses no initial estimate of g g g § 2.4 ). For linear models, Kong and Valiant [28 ] developed a method that approximates a quadratic form in the inverse of an unknown covariance matrix by polynomials, whose terms are estimated without bias by averages over distinct observations. Chen, Liu, and Mukherjee [10 Section 3.2] use Chebyshev polynomials of the reciprocal in this way. We apply this idea cell by cell, to the local Gram matrices of g g g
We first reduce to W = 0 W=0 W = 0 λ = 1 \lambda=1 λ = 1 ψ = ∫ a b g \psi=\int abg ψ = ∫ ab g T ^ \widehat{T} T T ( P ) = ∫ a b g T(P)=\int abg T ( P ) = ∫ ab g
∥ λ T ^ + 1 n ∑ i = 1 n W ( O i ) − ψ ( P ) ∥ L 2 ( P n ) ≤ ∣ λ ∣ ∥ T ^ − T ( P ) ∥ L 2 ( P n ) + ∥ W ∥ ∞ n − 1 / 2 . \left\lVert\lambda\widehat{T}+\frac{1}{n}\sum_{i=1}^{n}W(O_i)-\psi(P)\right\rVert_{L^2(P_n)}
\leq|\lambda|\left\lVert\widehat{T}-T(P)\right\rVert_{L^2(P_n)}+\lVert W\rVert_{\infty}n^{-1/2}. λ T + n 1 i = 1 ∑ n W ( O i ) − ψ ( P ) L 2 ( P n ) ≤ ∣ λ ∣ T − T ( P ) L 2 ( P n ) + ∥ W ∥ ∞ n − 1/2 . and the last term is at most a fixed multiple of both upper rates. We may also assume that α ≤ β \alpha\leq\beta α ≤ β ( a , U , α ) (a,U,\alpha) ( a , U , α ) ( b , V , β ) (b,V,\beta) ( b , V , β ) ∫ a b g \int abg ∫ ab g P ( α , β ) \mathcal{P}(\alpha,\beta) P ( α , β ) ( α , β ) (\alpha,\beta) ( α , β ) ℓ = ⌈ β ⌉ − 1 \ell=\lceil\beta\rceil-1 ℓ = ⌈ β ⌉ − 1 s 0 = min ( α , β ) s_0=\min(\alpha,\beta) s 0 = min ( α , β ) h ≤ 1 h\leq1 h ≤ 1 h s 0 h^{s_0} h s 0 h α h^\alpha h α h β h^\beta h β (10) that θ = ( α + β ) / d \theta=(\alpha+\beta)/d θ = ( α + β ) / d τ = − log ρ \tau=-\log\rho τ = − log ρ ν \nu ν (11) .
In this section, constants depend only on d , α , β , H , δ , g − , g + , M 0 d,\alpha,\beta,H,\delta,g_-,g_+,M_0 d , α , β , H , δ , g − , g + , M 0 ∣ λ ∣ |\lambda| ∣ λ ∣ ∥ W ∥ ∞ \lVert W\rVert_\infty ∥ W ∥ ∞ D , U , V , W D,U,V,W D , U , V , W 0 ≤ D ≤ M 0 0\leq D\leq M_0 0 ≤ D ≤ M 0 ∣ U ∣ , ∣ V ∣ ≤ M 0 |U|,|V|\leq M_0 ∣ U ∣ , ∣ V ∣ ≤ M 0 ∥ W ∥ ∞ \lVert W\rVert_\infty ∥ W ∥ ∞ π 0 \pi_0 π 0
The reciprocal series
Reciprocals enter both bounds. In the upper bound, we approximate products of reciprocals of cell moments of g g g 1 / x 1/x 1/ x I I I ρ I \rho_I ρ I ρ I − 1 \rho_I^{-1} ρ I − 1 x = 0 x=0 x = 0 [64 Chapter 8] . The Chebyshev coefficients of 1 / ( z − x ) 1/(z-x) 1/ ( z − x ) [42 Lemma 1, equation (10), p. 598] . For I = [ g − , g + ] I=[g_-,g_+] I = [ g − , g + ] ρ I \rho_I ρ I ρ \rho ρ (10) .
Lemma 4.1 (Reciprocal expansion). Fix an interval I = [ x − , x + ] I=[x_-,x_+] I = [ x − , x + ] 0 < x − < x + 0<x_-<x_+ 0 < x − < x +
c I = x + + x − 2 , d I = x + − x − 2 , g ˉ I = x − x + , ρ I = x + − x − x + + x − . c_I=\frac{x_++x_-}{2},\qquad d_I=\frac{x_+-x_-}{2},\qquad\bar{g}_I=\sqrt{x_-x_+},\qquad\rho_I=\frac{\sqrt{x_+}-\sqrt{x_-}}{\sqrt{x_+}+\sqrt{x_-}}. c I = 2 x + + x − , d I = 2 x + − x − , g ˉ I = x − x + , ρ I = x + + x − x + − x − . Let T k T_k T k T k ( cos t ) = cos ( k t ) T_k(\cos t)=\cos(kt) T k ( cos t ) = cos ( k t ) x ∈ I x\in I x ∈ I ∣ z ∣ < ρ I − 1 |z|<\rho_I^{-1} ∣ z ∣ < ρ I − 1
K I ( z ; x ) = 1 + 2 ∑ k ≥ 1 ( − ρ I ) k T k ( x − c I d I ) z k . K_I(z;x)=1+2\sum_{k\geq1}(-\rho_I)^kT_k\left(\frac{x-c_I}{d_I}\right)z^k. K I ( z ; x ) = 1 + 2 k ≥ 1 ∑ ( − ρ I ) k T k ( d I x − c I ) z k . Then
K I ( z ; x ) = 1 − ρ I 2 z 2 1 + 2 ρ I z ( x − c I ) / d I + ρ I 2 z 2 . (26) K_I(z;x)=\frac{1-\rho_I^2z^2}{1+2\rho_Iz(x-c_I)/d_I+\rho_I^2z^2}. \tag*{(26)} K I ( z ; x ) = 1 + 2 ρ I z ( x − c I ) / d I + ρ I 2 z 2 1 − ρ I 2 z 2 . (26) In particular, K I ( 1 ; x ) = g ˉ I / x K_I(1;x)=\bar{g}_I/x K I ( 1 ; x ) = g ˉ I / x x = c I + d I cos t x=c_I+d_I\cos t x = c I + d I cos t
1 c I + d I cos t = 1 g ˉ I { 1 + 2 ∑ k ≥ 1 ( − ρ I ) k cos ( k t ) } . (27) \frac{1}{c_I+d_I\cos t}=\frac{1}{\bar{g}_I}\left\{1+2\sum_{k\geq1}(-\rho_I)^k\cos(kt)\right\}. \tag*{(27)} c I + d I cos t 1 = g ˉ I 1 { 1 + 2 k ≥ 1 ∑ ( − ρ I ) k cos ( k t ) } . (27) Proof. We write x = c I + d I cos t x=c_I+d_I\cos t x = c I + d I cos t t t t q = − ρ I z q=-\rho_Iz q = − ρ I z T k ( ( x − c I ) / d I ) = cos ( k t ) T_k((x-c_I)/d_I)=\cos(kt) T k (( x − c I ) / d I ) = cos ( k t ) ∣ q ∣ < 1 |q|<1 ∣ q ∣ < 1 [61 Chapter 2]
1 + ∑ k ≥ 1 q k ( e i k t + e − i k t ) = 1 1 − q e i t + 1 1 − q e − i t − 1 = 1 − q 2 1 − 2 q cos t + q 2 , 1+\sum_{k\geq1}q^k\left(e^{ikt}+e^{-ikt}\right)=\frac{1}{1-qe^{it}}+\frac{1}{1-qe^{-it}}-1=\frac{1-q^2}{1-2q\cos t+q^2}, 1 + k ≥ 1 ∑ q k ( e ik t + e − ik t ) = 1 − q e i t 1 + 1 − q e − i t 1 − 1 = 1 − 2 q cos t + q 2 1 − q 2 , which is (4.1). At z = 1 z=1 z = 1 1 + ρ I 2 = 2 ρ I c I / d I 1+\rho_I^2=2\rho_Ic_I/d_I 1 + ρ I 2 = 2 ρ I c I / d I ( 1 − ρ I 2 ) d I / ( 2 ρ I ) = g ˉ I (1-\rho_I^2)d_I/(2\rho_I)=\bar{g}_I ( 1 − ρ I 2 ) d I / ( 2 ρ I ) = g ˉ I
For I = [ g − , g + ] I=[g_-,g_+] I = [ g − , g + ] ρ I = ρ \rho_I=\rho ρ I = ρ κ \kappa κ τ = − log ρ \tau=-\log\rho τ = − log ρ 1 / x = c I − 1 ∑ k ≥ 0 ( 1 − x / c I ) k 1/x=c_I^{-1}\sum_{k\geq0}(1-x/c_I)^k 1/ x = c I − 1 ∑ k ≥ 0 ( 1 − x / c I ) k I I I d I / c I = 2 ρ I / ( 1 + ρ I 2 ) d_I/c_I=2\rho_I/(1+\rho_I^2) d I / c I = 2 ρ I / ( 1 + ρ I 2 ) ρ I \rho_I ρ I − log { 2 ρ / ( 1 + ρ 2 ) } < τ -\log\{2\rho/(1+\rho^2)\}<\tau − log { 2 ρ / ( 1 + ρ 2 )} < τ τ \tau τ ρ I \rho_I ρ I 1 / x 1/x 1/ x I I I m m m ρ I m \rho_I^m ρ I m [29 Theorem 2.1 and Corollary 2.2] .
The HOIF estimators of Robins et al. with an estimated density expand the inverse of a Gram matrix about its value at an initial estimate [48 Theorems 3.14–3.15] . At resolutions with order n n n g g g [ g − , g + ] [g_{-},g_{+}] [ g − , g + ]
In the upper bound we use these series only on the interval [ g − , g + ] [g_{-},g_{+}] [ g − , g + ] c g = ( g − + g + ) / 2 c_{g}=(g_{-}+g_{+})/2 c g = ( g − + g + ) /2 d g = ( g + − g − ) / 2 d_{g}=(g_{+}-g_{-})/2 d g = ( g + − g − ) /2 g ˉ = g − g + \bar{g}=\sqrt{g_{-}g_{+}} g ˉ = g − g + c I , d I , g ˉ I c_{I},d_{I},\bar{g}_{I} c I , d I , g ˉ I ρ I = ρ \rho_{I}=\rho ρ I = ρ I I I 1 × 1 1\times1 1 × 1 M M M A M = ( M − c g I ) / d g A_{M}=(M-c_{g}I)/d_{g} A M = ( M − c g I ) / d g
Q ( z ; M ) = I + 2 ρ z A M + ρ 2 z 2 I , Q(z;M)=I+2\rho zA_{M}+\rho^{2}z^{2}I, Q ( z ; M ) = I + 2 ρ z A M + ρ 2 z 2 I , K ( z ; M ) = ( 1 − ρ 2 z 2 ) Q ( z ; M ) − 1 = I + 2 ∑ k ≥ 1 ( − ρ ) k T k ( A M ) z k . (28) \mathcal{K}(z;M)=(1-\rho^{2}z^{2})Q(z;M)^{-1}=I+2\sum_{k\geq1}(-\rho)^{k}T_{k}(A_{M})z^{k}. \tag*{(28)} K ( z ; M ) = ( 1 − ρ 2 z 2 ) Q ( z ; M ) − 1 = I + 2 k ≥ 1 ∑ ( − ρ ) k T k ( A M ) z k . (28) The last identity holds as a power series at z = 0 z=0 z = 0 T k + 1 ( A ) = 2 A T k ( A ) − T k − 1 ( A ) T_{k+1}(A)=2AT_{k}(A)-T_{k-1}(A) T k + 1 ( A ) = 2 A T k ( A ) − T k − 1 ( A ) T 0 ( A ) = I T_{0}(A)=I T 0 ( A ) = I T 1 ( A ) = A T_{1}(A)=A T 1 ( A ) = A Q ( z ; M ) Q(z;M) Q ( z ; M ) ( 1 − ρ 2 z 2 ) I (1-\rho^{2}z^{2})I ( 1 − ρ 2 z 2 ) I Q ( 0 ; M ) = I Q(0;M)=I Q ( 0 ; M ) = I z k z^{k} z k k k k M M M M M M [ g − , g + ] [g_{-},g_{+}] [ g − , g + ] A M A_{M} A M [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] ∥ T k ( A M ) ∥ o p ≤ 1 \lVert T_{k}(A_{M})\rVert_{\mathrm{op}}\leq1 ∥ T k ( A M ) ∥ op ≤ 1 Lemma 4.1 , we obtain absolute convergence of (28) for ∣ z ∣ < ρ − 1 |z|<\rho^{-1} ∣ z ∣ < ρ − 1 K ( 1 ; M ) = g ˉ M − 1 \mathcal{K}(1;M)=\bar{g}M^{-1} K ( 1 ; M ) = g ˉ M − 1
We expand all reciprocal factors in the same variable z z z m m m Lemma 4.5 , the derivatives uniformly in the degree. For a function f f f C p \mathbb{C}^{p} C p t 0 t_{0} t 0 v 1 , … , v r ∈ R p v_{1},\ldots,v_{r}\in\mathbb{R}^{p} v 1 , … , v r ∈ R p D r f ( t 0 ) [ v 1 , … , v r ] D^{r}f(t_{0})[v_{1},\ldots,v_{r}] D r f ( t 0 ) [ v 1 , … , v r ] ∂ s 1 ⋯ ∂ s r f ( t 0 + s 1 v 1 + ⋯ + s r v r ) \partial_{s_{1}}\cdots\partial_{s_{r}}f(t_{0}+s_{1}v_{1}+\cdots+s_{r}v_{r}) ∂ s 1 ⋯ ∂ s r f ( t 0 + s 1 v 1 + ⋯ + s r v r ) s = 0 s=0 s = 0
Lemma 4.2 (Reciprocal truncations and derivative bounds). (a) Fix integers N ≥ 0 N\geq0 N ≥ 0 p 0 ≥ 1 p_{0}\geq1 p 0 ≥ 1 p 1 , … , p N ≥ 0 p_{1},\ldots,p_{N}\geq0 p 1 , … , p N ≥ 0 ν ′ = p 1 + ⋯ + p N \nu'=p_{1}+\cdots+p_{N} ν ′ = p 1 + ⋯ + p N M 0 , … , M N M_{0},\ldots,M_{N} M 0 , … , M N p 0 , … , p N p_{0},\ldots,p_{N} p 0 , … , p N m ≥ 0 m\geq0 m ≥ 0 G k G_{k} G k S m S_{m} S m z z z
K ( z ; M 0 ) ∏ i = 1 N det K ( z ; M i ) = ∑ k ≥ 0 G k z k , S m ( M 0 , … , M N ) = g ˉ − ( ν ′ + 1 ) ∑ k = 0 m G k , (29) \mathcal{K}(z;M_{0})\prod_{i=1}^{N}\det\mathcal{K}(z;M_{i})=\sum_{k\geq0}G_{k}z^{k},\qquad S_{m}(M_{0},\ldots,M_{N})=\bar{g}^{-(\nu'+1)}\sum_{k=0}^{m}G_{k}, \tag*{(29)} K ( z ; M 0 ) i = 1 ∏ N det K ( z ; M i ) = k ≥ 0 ∑ G k z k , S m ( M 0 , … , M N ) = g ˉ − ( ν ′ + 1 ) k = 0 ∑ m G k , (29) where a determinant of dimension zero equals one. Then S m S_{m} S m m m m M 0 , … , M N M_{0},\ldots,M_{N} M 0 , … , M N M 0 , … , M N M_{0},\ldots,M_{N} M 0 , … , M N [ g − , g + ] [g_{-},g_{+}] [ g − , g + ]
∥ M 0 − 1 det M 1 ⋯ det M N − S m ( M 0 , … , M N ) ∥ o p ≤ C ( m + 1 ) ν ′ ρ m , (30) \left\lVert\frac{M_{0}^{-1}}{\det M_{1}\cdots\det M_{N}}-S_{m}(M_{0},\ldots,M_{N})\right\rVert_{\mathrm{op}}\leq C(m+1)^{\nu'}\rho^{m}, \tag*{(30)} det M 1 ⋯ det M N M 0 − 1 − S m ( M 0 , … , M N ) op ≤ C ( m + 1 ) ν ′ ρ m , (30) C = 2 ν ′ + 1 g ˉ − ( ν ′ + 1 ) ∑ k ≥ 0 ( k + 1 ) ν ′ ρ k . C=2^{\nu'+1}\bar{g}^{-(\nu'+1)}\sum_{k\geq0}(k+1)^{\nu'}\rho^{k}. C = 2 ν ′ + 1 g ˉ − ( ν ′ + 1 ) k ≥ 0 ∑ ( k + 1 ) ν ′ ρ k . (b) Let V ⊂ R p \mathcal{V} \subset\mathbb{R}^{p} V ⊂ R p M 1 ( t ) , … , M N ( t ) M_1(t),\ldots,M_N(t) M 1 ( t ) , … , M N ( t ) C p \mathbb{C}^{p} C p [ g − , g + ] [g_{-},g_{+}] [ g − , g + ] t ∈ V t \in\mathcal{V} t ∈ V S ( z ; t ) = ∑ k ≥ 0 S k ( t ) z k S(z;t)=\sum_{k\geq0}S_k(t)z^k S ( z ; t ) = ∑ k ≥ 0 S k ( t ) z k K ( z ; M i ( t ) ) \mathcal{K}(z;M_i(t)) K ( z ; M i ( t )) 1 ≤ i ≤ N 1\leq i\leq N 1 ≤ i ≤ N t t t ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ] C C C ∣ ∑ k = 0 m S k ( t ) ∣ ≤ C \left|\sum_{k=0}^{m}S_k(t)\right|\leq C ∣ ∑ k = 0 m S k ( t ) ∣ ≤ C m ≥ 0 m\geq0 m ≥ 0 t ∈ C p t\in\mathbb{C}^{p} t ∈ C p ∥ t − t 0 ∥ ∞ < ε \lVert t-t_0\rVert_{\infty}<\varepsilon ∥ t − t 0 ∥ ∞ < ε t 0 ∈ V t_0\in\mathcal{V} t 0 ∈ V g ± g_{\pm} g ± M i M_i M i S S S V \mathcal{V} V
(c) Let f f f C p \mathbb{C}^{p} C p t 0 ∈ R p t_0\in\mathbb{R}^{p} t 0 ∈ R p ε > 0 \varepsilon>0 ε > 0 ∣ f ( t ) ∣ ≤ A |f(t)|\leq A ∣ f ( t ) ∣ ≤ A ∥ t − t 0 ∥ ∞ < ε \lVert t-t_0\rVert_{\infty}<\varepsilon ∥ t − t 0 ∥ ∞ < ε r ≥ 1 r\geq1 r ≥ 1 v 1 , … , v r ∈ R p v_1,\ldots,v_r\in\mathbb{R}^{p} v 1 , … , v r ∈ R p
∣ D r f ( t 0 ) [ v 1 , … , v r ] ∣ ≤ A r ! ( e / ε ) r ∏ i = 1 r ∥ v i ∥ ∞ . \left|D^r f(t_0)[v_1,\ldots,v_r]\right|\leq A r!(e/\varepsilon)^r\prod_{i=1}^{r}\lVert v_i\rVert_{\infty}. ∣ D r f ( t 0 ) [ v 1 , … , v r ] ∣ ≤ A r ! ( e / ε ) r i = 1 ∏ r ∥ v i ∥ ∞ . Proof. (a) The coefficient of z k z^k z k K ( z ; M ) \mathcal{K}(z;M) K ( z ; M ) k k k M M M z k z^k z k det K ( z ; M i ) \det\mathcal{K}(z;M_i) det K ( z ; M i ) k k k G k G_k G k S m S_m S m m m m [ g − , g + ] [g_{-},g_{+}] [ g − , g + ] M i M_i M i det K ( z ; M i ) = ∏ l = 1 p i K ( z ; λ i l ) \det\mathcal{K}(z;M_i)=\prod_{l=1}^{p_i}\mathcal{K}(z;\lambda_{il}) det K ( z ; M i ) = ∏ l = 1 p i K ( z ; λ i l ) ∣ z ∣ < ρ − 1 |z|<\rho^{-1} ∣ z ∣ < ρ − 1 λ i l \lambda_{il} λ i l M i M_i M i (29) is therefore a product of ν ′ + 1 \nu'+1 ν ′ + 1 z k z^k z k 2 ρ k 2^{\rho k} 2 ρ k ( k + ν ′ ν ′ ) ≤ ( k + 1 ) ν ′ \binom{k+\nu'}{\nu'}\leq(k+1)^{\nu'} ( ν ′ k + ν ′ ) ≤ ( k + 1 ) ν ′ k k k ν ′ + 1 \nu'+1 ν ′ + 1
∥ G k ∥ op ≤ 2 ν ′ + 1 ( k + 1 ) ν ′ ρ k . \lVert G_k\rVert_{\operatorname{op}}\leq2^{\nu'+1}(k+1)^{\nu'}\rho^k. ∥ G k ∥ op ≤ 2 ν ′ + 1 ( k + 1 ) ν ′ ρ k . In particular, ∑ k G k \sum_kG_k ∑ k G k K ( 1 ; M ) = g ˉ M − 1 \mathcal{K}(1;M)=\bar{g}M^{-1} K ( 1 ; M ) = g ˉ M − 1 det ( g ˉ M i − 1 ) = g ˉ p i / det M i \det(\bar{g}M_i^{-1})=\bar{g}^{p_i}/\det M_i det ( g ˉ M i − 1 ) = g ˉ p i / det M i g ˉ ν ′ + 1 M 0 − 1 / ( det M 1 ⋯ det M N ) \bar{g}^{\nu'+1}M_0^{-1}/(\det M_1\cdots\det M_N) g ˉ ν ′ + 1 M 0 − 1 / ( det M 1 ⋯ det M N ) k = m + j k=m+j k = m + j j ≥ 1 j\geq1 j ≥ 1 m + j + 1 ≤ ( m + 1 ) ( j + 1 ) m+j+1\leq(m+1)(j+1) m + j + 1 ≤ ( m + 1 ) ( j + 1 ) (30) by
g ˉ − ( ν ′ + 1 ) ∑ k > m ∥ G k ∥ op ≤ 2 ν ′ + 1 g ˉ − ( ν ′ + 1 ) ( m + 1 ) ν ′ ρ m ∑ j ≥ 1 ( j + 1 ) ν ′ ρ j , \bar{g}^{-(\nu'+1)}\sum_{k>m}\lVert G_k\rVert_{\operatorname{op}}
\leq2^{\nu'+1}\bar{g}^{-(\nu'+1)}(m+1)^{\nu'}\rho^m\sum_{j\geq1}(j+1)^{\nu'}\rho^j, g ˉ − ( ν ′ + 1 ) k > m ∑ ∥ G k ∥ op ≤ 2 ν ′ + 1 g ˉ − ( ν ′ + 1 ) ( m + 1 ) ν ′ ρ m j ≥ 1 ∑ ( j + 1 ) ν ′ ρ j , which is at most the right side of (30) .
(b) Let r ∗ = ρ − 1 / 2 r_*=\rho^{-1/2} r ∗ = ρ − 1/2 b = ( 1 − ρ ) 2 b=(1-\sqrt{\rho})^2 b = ( 1 − ρ ) 2 M M M [ g − , g + ] [g_{-},g_{+}] [ g − , g + ] Q ( z ; M ) Q(z;M) Q ( z ; M ) M M M ( 1 + ρ z e i ϕ ) ( 1 + ρ z e − i ϕ ) (1+\rho z e^{i\phi})(1+\rho z e^{-i\phi}) ( 1 + ρ z e i ϕ ) ( 1 + ρ z e − i ϕ ) ϕ \phi ϕ ( 1 − ρ r ∗ ) 2 = b (1-\rho r_*)^2=b ( 1 − ρ r ∗ ) 2 = b ∣ z ∣ ≤ r ∗ |z|\leq r_* ∣ z ∣ ≤ r ∗ ∥ Q ( z ; M ) − 1 ∥ op ≤ b − 1 \lVert Q(z;M)^{-1}\rVert_{\operatorname{op}}\leq b^{-1} ∥ Q ( z ; M ) − 1 ∥ op ≤ b − 1 ∣ z ∣ ≤ r ∗ |z|\leq r_* ∣ z ∣ ≤ r ∗ Λ ≥ 1 \Lambda\geq1 Λ ≥ 1 ∥ M i ( t ) − M i ( t 0 ) ∥ op ≤ Λ ∥ t − t 0 ∥ ∞ \lVert M_i(t)-M_i(t_0)\rVert_{\operatorname{op}}\leq\Lambda\lVert t-t_0\rVert_{\infty} ∥ M i ( t ) − M i ( t 0 ) ∥ op ≤ Λ ∥ t − t 0 ∥ ∞ i i i t , t 0 t,t_0 t , t 0 ε = min { 1 , d g b / ( 4 Λ ) } \varepsilon=\min\{1,d_gb/(4\Lambda)\} ε = min { 1 , d g b / ( 4Λ )} t 0 ∈ V t_0\in\mathcal{V} t 0 ∈ V ∥ t − t 0 ∥ ∞ < ε \lVert t-t_0\rVert_{\infty}<\varepsilon ∥ t − t 0 ∥ ∞ < ε
∥ Q ( z ; M i ( t ) ) − Q ( z ; M i ( t 0 ) ) ∥ op ≤ 2 ρ r ∗ Λ ε / d g ≤ b / 2 , ∣ z ∣ ≤ r ∗ , \lVert Q(z;M_i(t))-Q(z;M_i(t_0))\rVert_{\operatorname{op}}\leq2\rho r_*\Lambda\varepsilon/d_g\leq b/2,\qquad|z|\leq r_*, ∥ Q ( z ; M i ( t )) − Q ( z ; M i ( t 0 )) ∥ op ≤ 2 ρ r ∗ Λ ε / d g ≤ b /2 , ∣ z ∣ ≤ r ∗ , and the Neumann series shows that ∥ Q ( z ; M i ( t ) ) − 1 ∥ op ≤ 2 / b \lVert Q(z;M_i(t))^{-1}\rVert_{\operatorname{op}}\leq2/b ∥ Q ( z ; M i ( t ) ) − 1 ∥ op ≤ 2/ b ∥ K ( z ; M i ( t ) ) ∥ op ≤ 2 ( 1 + ρ ) / b \lVert\mathcal{K}(z;M_i(t))\rVert_{\operatorname{op}}\leq2(1+\rho)/b ∥ K ( z ; M i ( t )) ∥ op ≤ 2 ( 1 + ρ ) / b ∥ t − t 0 ∥ ∞ < ε \lVert t-t_0\rVert_{\infty}<\varepsilon ∥ t − t 0 ∥ ∞ < ε ∣ z ∣ ≤ r ∗ |z|\leq r_* ∣ z ∣ ≤ r ∗ t t t S ( ⋅ ; t ) S(\cdot;t) S ( ⋅ ; t ) A A A ∣ z ∣ ≤ r ∗ \lvert z\rvert\le r_* ∣ z ∣ ≤ r ∗ S k ( t ) S_k(t) S k ( t ) (28) ; by Cauchy’s inequality, we have ∣ S k ( t ) ∣ ≤ A r ∗ − k \lvert S_k(t)\rvert\le Ar_*^{-k} ∣ S k ( t )∣ ≤ A r ∗ − k k ≤ m k \le m k ≤ m A / ( 1 − ρ ) A/(1-\sqrt{\rho}) A / ( 1 − ρ )
(c) By multilinearity we may assume that ∥ v i ∥ ∞ = 1 \lVert v_i\rVert_\infty=1 ∥ v i ∥ ∞ = 1 i i i φ ( w 1 , … , w r ) = f ( t 0 + ∑ i w i v i ) \varphi(w_1,\ldots,w_r)=f(t_0+\sum_i w_i v_i) φ ( w 1 , … , w r ) = f ( t 0 + ∑ i w i v i ) C r \mathbb{C}^r C r A A A max i ∣ w i ∣ < ε / r \max_i\lvert w_i\rvert<\varepsilon/r max i ∣ w i ∣ < ε / r
D r f ( t 0 ) [ v 1 , … , v r ] = ∂ w 1 ⋯ ∂ w r φ ( 0 ) . D^r f(t_0)[v_1,\ldots,v_r]=\partial_{w_1}\cdots\partial_{w_r}\varphi(0). D r f ( t 0 ) [ v 1 , … , v r ] = ∂ w 1 ⋯ ∂ w r φ ( 0 ) . By Cauchy’s formula on the torus ∣ w 1 ∣ = ⋯ = ∣ w r ∣ = ε ′ \lvert w_1\rvert=\cdots=\lvert w_r\rvert=\varepsilon' ∣ w 1 ∣ = ⋯ = ∣ w r ∣ = ε ′ ε ′ < ε / r \varepsilon'<\varepsilon/r ε ′ < ε / r A ε ′ − r A\varepsilon'^{-r} A ε ′ − r ε ′ ↑ ε / r \varepsilon'\uparrow\varepsilon/r ε ′ ↑ ε / r r r ≤ e r r ! r^r\le e^r r! r r ≤ e r r !
Projection increments and their polynomial approximations
We bisect [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d 1 , … , d 1,\ldots,d 1 , … , d j j j C j \mathcal{C}_j C j 2 j 2^j 2 j V j V_j V j C j \mathcal{C}_j C j ℓ \ell ℓ Π j \Pi_j Π j V j V_j V j L 2 ( g ) L^2(g) L 2 ( g ) [1 Section 1.1] . Haar projections of this kind were combined with U-statistics to estimate integral functionals of a density in [27 ] . We define ψ j = ⟨ Π j a , Π j b ⟩ g \psi_j=\langle\Pi_j a,\Pi_j b\rangle_g ψ j = ⟨ Π j a , Π j b ⟩ g ⟨ f 1 , f 2 ⟩ g = ∫ f 1 f 2 g \langle f_1,f_2\rangle_g=\int f_1f_2g ⟨ f 1 , f 2 ⟩ g = ∫ f 1 f 2 g
ψ − ψ j = ⟨ a − Π j a , b − Π j b ⟩ g , ∣ ψ − ψ j ∣ ≤ C 2 − j θ . (31) \psi-\psi_j=\langle a-\Pi_j a,b-\Pi_j b\rangle_g,\qquad\lvert\psi-\psi_j\rvert\le C2^{-j\theta}. \tag*{(31)} ψ − ψ j = ⟨ a − Π j a , b − Π j b ⟩ g , ∣ ψ − ψ j ∣ ≤ C 2 − j θ . (31) Indeed, the cellwise Taylor polynomials of degrees ⌈ α ⌉ − 1 \lceil\alpha\rceil-1 ⌈ α ⌉ − 1 ⌈ β ⌉ − 1 \lceil\beta\rceil-1 ⌈ β ⌉ − 1 V j V_j V j C 2 − j α / d C2^{-j\alpha/d} C 2 − j α / d C 2 − j β / d C2^{-j\beta/d} C 2 − j β / d L 2 ( g ) L^2(g) L 2 ( g ) g + g_+ g + (31) has the form of the truncation bias of the HOIF estimators, which is the expectation of a product of two projection residuals [48 Theorems 3.7–3.8] .
The decomposition ψ J = ψ 0 + ∑ j = 1 J ( ψ j − ψ j − 1 ) \psi_J=\psi_0+\sum_{j=1}^J(\psi_j-\psi_{j-1}) ψ J = ψ 0 + ∑ j = 1 J ( ψ j − ψ j − 1 ) g g g a g ag a g b g bg b g
Lemma 4.3 (Polynomials for projection increments). Fix a level j ≥ 1 j\ge1 j ≥ 1 K = 2 j − 1 K=2^{j-1} K = 2 j − 1 h = K − 1 / d h=K^{-1/d} h = K − 1/ d C ∈ C j − 1 C\in\mathcal{C}_{j-1} C ∈ C j − 1 Z C ( O ) Z_C(O) Z C ( O ) m C = E P Z C ( O ) m_C=\mathbb{E}_P Z_C(O) m C = E P Z C ( O ) F C F_C F C
ψ j − ψ j − 1 = K − 1 ∑ C ∈ C j − 1 F C ( m C ) , ∣ F C ( m C ) ∣ ≤ C 0 h α + β , \psi_j-\psi_{j-1}=K^{-1}\sum_{C\in\mathcal{C}_{j-1}}F_C(m_C),\qquad\lvert F_C(m_C)\rvert\le C_0h^{\alpha+\beta}, ψ j − ψ j − 1 = K − 1 C ∈ C j − 1 ∑ F C ( m C ) , ∣ F C ( m C )∣ ≤ C 0 h α + β , and
∥ Z C ( O ) ∥ 2 ≤ C 0 K 1 C ( X ) , P ( X ∈ C ) ≤ C 0 / K . (32) \lVert Z_C(O)\rVert_2 \le C_0K1_C(X), \qquad P(X \in C) \le C_0/K. \tag*{(32)} ∥ Z C ( O ) ∥ 2 ≤ C 0 K 1 C ( X ) , P ( X ∈ C ) ≤ C 0 / K . (32) For every integer m ≥ 0 m \ge0 m ≥ 0 F C , m F_{C,m} F C , m m + ν + 2 m+\nu+2 m + ν + 2
∣ F C ( m C ) − F C , m ( m C ) ∣ ≤ C 0 h α + β ( m + 1 ) ν ρ m . (33) \lvert F_C(\mathbf{m}_C)-F_{C,m}(\mathbf{m}_C)\rvert\le C_0h^{\alpha+\beta}(m+1)^\nu\rho^m. \tag*{(33)} ∣ F C ( m C ) − F C , m ( m C )∣ ≤ C 0 h α + β ( m + 1 ) ν ρ m . (33) At every population moment vector, that is, at m C = m C ( P ) \mathbf{m}_C=\mathbf{m}_C(P) m C = m C ( P ) P ∈ P ( α , β ) P \in\mathcal{P}(\alpha,\beta) P ∈ P ( α , β )
∣ F C , m ( t ) ∣ ≤ C 0 for complex t with ∥ t − m C ∥ ∞ < 1 / C 0 , (34) \lvert F_{C,m}(\mathbf{t})\rvert\le C_0 \qquad\text{for complex }\mathbf{t}\text{ with }\lVert\mathbf{t}-\mathbf{m}_C\rVert_\infty< 1/C_0, \tag*{(34)} ∣ F C , m ( t )∣ ≤ C 0 for complex t with ∥ t − m C ∥ ∞ < 1/ C 0 , (34) ∥ ∇ F C , m ( m C ) ∥ 2 ≤ C 0 h s 0 . (35) \lVert\nabla F_{C,m}(\mathbf{m}_C)\rVert_2 \le C_0h^{s_0}. \tag*{(35)} ∥ ∇ F C , m ( m C ) ∥ 2 ≤ C 0 h s 0 . (35) The functions Z C Z_C Z C F C , m F_{C,m} F C , m m m m Z C Z_C Z C C 0 ≥ 1 C_0 \ge1 C 0 ≥ 1 j j j m m m C C C P P P Z 0 ( O ) Z_0(O) Z 0 ( O ) m ≥ 0 m \ge0 m ≥ 0 F 0 , m F_{0,m} F 0 , m m + 2 m+2 m + 2 m 0 = E P Z 0 ( O ) \mathbf{m}_0=\mathbb{E}_P Z_0(O) m 0 = E P Z 0 ( O ) ∣ ψ 0 − F 0 , m ( m 0 ) ∣ ≤ C 0 ρ m \lvert\psi_0-F_{0,m}(\mathbf{m}_0)\rvert\le C_0\rho^m ∣ ψ 0 − F 0 , m ( m 0 )∣ ≤ C 0 ρ m K = h = 1 K=h=1 K = h = 1
Proof. Bases. For an increment, let C ∈ C j − 1 C \in\mathcal{C}_{j-1} C ∈ C j − 1 1 / K 1/K 1/ K 2 d h 2\sqrt{d}h 2 d h C C C ℓ \ell ℓ η = a , b \eta=a,b η = a , b η \eta η C C C ⌈ α ⌉ − 1 \lceil\alpha\rceil-1 ⌈ α ⌉ − 1 η = a \eta=a η = a ℓ \ell ℓ η = b \eta=b η = b p a = q α p_a=q_\alpha p a = q α p b = q β p_b=q_\beta p b = q β α ≤ β \alpha\le\beta α ≤ β a a a b b b C = [ 0 , 1 ] d C=[0,1]^d C = [ 0 , 1 ] d K = h = 1 K=h=1 K = h = 1 ℓ \ell ℓ { 0 } \{0\} { 0 } p a = p b = 0 p_a=p_b=0 p a = p b = 0
Let z C z_C z C K d x K\,dx K d x C C C η = a , b \eta=a,b η = a , b L η ⊤ z C L_\eta^\top z_C L η ⊤ z C η \eta η K d x K\,dx K d x C C C L η L_\eta L η p η p_\eta p η L η ⊤ L η = I L_\eta^\top L_\eta=I L η ⊤ L η = I L a L_a L a L b L_b L b d d d [ 0 , 1 2 ] r × [ 0 , 1 ] d − r [0,\frac{1}{2}]^r \times[0,1]^{d-r} [ 0 , 2 1 ] r × [ 0 , 1 ] d − r 0 ≤ r < d 0 \le r < d 0 ≤ r < d r + 1 r+1 r + 1 [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d z C z_C z C L a , L b L_a,L_b L a , L b
Moments. Suppressing the cell index, we define
Ω = K E P [ 1 C ( X ) D z C ( X ) z C ( X ) ⊤ ] , Γ η = L η ⊤ Ω L η , u = K E P [ 1 C ( X ) U z C ( X ) ] , v = K E P [ 1 C ( X ) V z C ( X ) ] . (36) \begin{aligned}
\Omega&= K\mathbb{E}_P[1_C(X)Dz_C(X)z_C(X)^\top], \qquad\Gamma_\eta=L_\eta^\top\Omega L_\eta,\\
u &= K\mathbb{E}_P[1_C(X)Uz_C(X)], \qquad v=K\mathbb{E}_P[1_C(X)Vz_C(X)]. \tag*{(36)}
\end{aligned} Ω u = K E P [ 1 C ( X ) D z C ( X ) z C ( X ) ⊤ ] , Γ η = L η ⊤ Ω L η , = K E P [ 1 C ( X ) U z C ( X )] , v = K E P [ 1 C ( X ) V z C ( X )] . (36) They equal K ∫ C g z C z C ⊤ K\int_C gz_Cz_C^\top K ∫ C g z C z C ⊤ K ∫ C a g z C K\int_C agz_C K ∫ C a g z C K ∫ C b g z C K\int_C bgz_C K ∫ C b g z C z C z_C z C K d x K\,dx K d x C C C g − ≤ g ≤ g + g_- \leq g \leq g_+ g − ≤ g ≤ g + g − I ⪯ Ω ⪯ g + I g_-I \preceq\Omega\preceq g_+I g − I ⪯ Ω ⪯ g + I L η ⊤ L η = I L_\eta^\top L_\eta=I L η ⊤ L η = I Γ η \Gamma_\eta Γ η p η ≥ 1 p_\eta\geq1 p η ≥ 1 f f f C C C
∥ K ∫ C f g z C ∥ 2 ≤ ( K ∫ C f 2 g 2 ) 1 / 2 ≤ g + sup C ∣ f ∣ . (37) \left\|K\int_C fgz_C\right\|_2 \leq\left(K\int_C f^2g^2\right)^{1/2} \leq g_+\sup_C |f|. \tag*{(37)} K ∫ C f g z C 2 ≤ ( K ∫ C f 2 g 2 ) 1/2 ≤ g + C sup ∣ f ∣. (37) With f = a , b f=a,b f = a , b ∥ u ∥ 2 , ∥ v ∥ 2 ≤ H g + \|u\|_2,\|v\|_2 \leq Hg_+ ∥ u ∥ 2 , ∥ v ∥ 2 ≤ H g + m C m_C m C Ω \Omega Ω u u u v v v m C = E P Z C ( O ) m_C=E_PZ_C(O) m C = E P Z C ( O ) Z C ( O ) Z_C(O) Z C ( O ) K 1 C ( X ) D z C ( X ) z C ( X ) ⊤ K1_C(X)D_{z_C}(X)z_C(X)^\top K 1 C ( X ) D z C ( X ) z C ( X ) ⊤ K 1 C ( X ) U z C ( X ) K1_C(X)Uz_C(X) K 1 C ( X ) U z C ( X ) K 1 C ( X ) V z C ( X ) K1_C(X)Vz_C(X) K 1 C ( X ) V z C ( X ) ∥ Z C ( O ) ∥ 2 ≤ C 0 K 1 C ( X ) \|Z_C(O)\|_2\leq C_0K1_C(X) ∥ Z C ( O ) ∥ 2 ≤ C 0 K 1 C ( X ) p = g / w ≤ g + / δ p=g/w\leq g_+/\delta p = g / w ≤ g + / δ P ( X ∈ C ) ≤ C 0 / K P(X\in C)\leq C_0/K P ( X ∈ C ) ≤ C 0 / K
The increment. We set B η = L η Γ η − 1 L η ⊤ B_\eta=L_\eta\Gamma_\eta^{-1}L_\eta^\top B η = L η Γ η − 1 L η ⊤ N η = Ω − 1 − B η N_\eta=\Omega^{-1}-B_\eta N η = Ω − 1 − B η B η = 0 B_\eta=0 B η = 0 det Γ η = 1 \det\Gamma_\eta=1 det Γ η = 1 η \eta η { 0 } \{0\} { 0 } ⟨ f 1 , f 2 ⟩ g , C = K ∫ C f 1 f 2 g \langle f_1,f_2\rangle_{g,C}=K\int_C f_1f_2g ⟨ f 1 , f 2 ⟩ g , C = K ∫ C f 1 f 2 g z C z_C z C L η ⊤ z C L_\eta^\top z_C L η ⊤ z C Ω \Omega Ω Γ η \Gamma_\eta Γ η z C z_C z C a a a b b b u u u v v v a a a η \eta η Ω − 1 u \Omega^{-1}u Ω − 1 u B η u B_\eta u B η u z C z_C z C N η u N_\eta u N η u E η a E_\eta a E η a E η E_\eta E η η \eta η ⟨ ⋅ , ⋅ ⟩ g , C \langle\cdot,\cdot\rangle_{g,C} ⟨ ⋅ , ⋅ ⟩ g , C b b b v v v Π j \Pi_j Π j Π j − 1 \Pi_{j-1} Π j − 1 C C C b b b K K K C C C ψ j − ψ j − 1 \psi_j-\psi_{j-1} ψ j − ψ j − 1
F C = u ⊤ Ω − 1 v − u ⊤ B b v = u ⊤ N b v . F_C=u^\top\Omega^{-1}v-u^\top B_bv=u^\top N_bv. F C = u ⊤ Ω − 1 v − u ⊤ B b v = u ⊤ N b v . For the base term, the same formula gives F C = u ⊤ Ω − 1 v = ψ 0 F_C=u^\top\Omega^{-1}v=\psi_0 F C = u ⊤ Ω − 1 v = ψ 0 [48 p. 357] , we represent a projection through the inverse of a Gram matrix; here the Gram matrices are local, one for each parent cell.
Two small polynomial factors. Both terms of F C F_C F C x ⊤ Ω y x^\top\Omega y x ⊤ Ω y B η Ω B_\eta\Omega B η Ω L η L_\eta L η N η Ω = I − B η Ω N_\eta\Omega=I-B_\eta\Omega N η Ω = I − B η Ω L a L_a L a L b L_b L b ( N a Ω ) ( N b Ω ) = N b Ω (N_a\Omega)(N_b\Omega)=N_b\Omega ( N a Ω ) ( N b Ω ) = N b Ω N a Ω N b = N b N_a\Omega N_b=N_b N a Ω N b = N b
R η = ( det Γ η ) I − Ω L η adj ( Γ η ) L η ⊤ = ( det Γ η ) Ω N η . R_\eta=(\det\Gamma_\eta)I-\Omega L_\eta\operatorname{adj}(\Gamma_\eta)L_\eta^\top=(\det\Gamma_\eta)\Omega N_\eta. R η = ( det Γ η ) I − Ω L η adj ( Γ η ) L η ⊤ = ( det Γ η ) Ω N η . where adj ( Γ η ) = ( det Γ η ) Γ η − 1 \operatorname{adj}(\Gamma_\eta)=(\det\Gamma_\eta)\Gamma_\eta^{-1} adj ( Γ η ) = ( det Γ η ) Γ η − 1 R η = I R_\eta=I R η = I η \eta η { 0 } \{0\} { 0 }
r a = R a u , r b = R b v , S = Ω − 1 det Γ a det Γ b . (38) r_a=R_au,\qquad r_b=R_bv,\qquad S=\frac{\Omega^{-1}}{\det\Gamma_a\det\Gamma_b}. \tag*{(38)} r a = R a u , r b = R b v , S = det Γ a det Γ b Ω − 1 . (38) Since N a N_a N a Ω \Omega Ω R a ⊤ S R b = N a Ω Ω − 1 Ω N b = N b R_a^\top SR_b=N_a\Omega\Omega^{-1}\Omega N_b=N_b R a ⊤ S R b = N a Ω Ω − 1 Ω N b = N b F C = r a ⊤ S r b F_C=r_a^\top Sr_b F C = r a ⊤ S r b det Γ η \det\Gamma_\eta det Γ η adj ( Γ η ) \operatorname{adj}(\Gamma_\eta) adj ( Γ η ) p η p_\eta p η p η − 1 p_\eta-1 p η − 1 Ω \Omega Ω r a r_a r a r b r_b r b m C m_C m C p a + 1 p_a+1 p a + 1 p b + 1 p_b+1 p b + 1
To see that r a r_a r a a ~ \tilde{a} a ~ a a a ⌈ α ⌉ − 1 \lceil\alpha\rceil-1 ⌈ α ⌉ − 1 C C C ∣ a − a ~ ∣ ≤ C h α \lvert a-\tilde{a}\rvert\le Ch^\alpha ∣ a − a ~ ∣ ≤ C h α C C C a ~ = 0 \tilde{a}=0 a ~ = 0 ∣ a − a ~ ∣ ≤ H \lvert a-\tilde{a}\rvert\le H ∣ a − a ~ ∣ ≤ H a ~ \tilde{a} a ~ a a a E a a ~ = 0 E_a\tilde{a}=0 E a a ~ = 0 ∥ x ∥ Ω 2 = x ⊤ Ω x \lVert x\rVert_\Omega^2=x^\top\Omega x ∥ x ∥ Ω 2 = x ⊤ Ω x Ω 2 ⪯ g + Ω \Omega^2\preceq g_+\Omega Ω 2 ⪯ g + Ω ∥ Ω x ∥ 2 ≤ g + 1 / 2 ∥ x ∥ Ω \lVert\Omega x\rVert_2\le g_+^{1/2}\lVert x\rVert_\Omega ∥ Ω x ∥ 2 ≤ g + 1/2 ∥ x ∥ Ω det Γ a ≤ g + p a \det\Gamma_a\le g_+^{p_a} det Γ a ≤ g + p a E a E_a E a
∥ r a ∥ 2 ≤ g + p a + 1 / 2 ∥ N a u ∥ Ω = g + p a + 1 / 2 ∥ E a ( a − a ~ ) ∥ g , C ≤ g + p a + 1 sup C ∣ a − a ~ ∣ ≤ C h α . \lVert r_a\rVert_2\le g_+^{p_a+1/2}\lVert N_a u\rVert_\Omega=g_+^{p_a+1/2}\lVert E_a(a-\tilde{a})\rVert_{g,C}\le g_+^{p_a+1}\sup_C\lvert a-\tilde{a}\rvert\le Ch^\alpha. ∥ r a ∥ 2 ≤ g + p a + 1/2 ∥ N a u ∥ Ω = g + p a + 1/2 ∥ E a ( a − a ~ ) ∥ g , C ≤ g + p a + 1 C sup ∣ a − a ~ ∣ ≤ C h α . where ∥ f ∥ g , C 2 = ⟨ f , f ⟩ g , C \lVert f\rVert_{g,C}^2=\langle f,f\rangle_{g,C} ∥ f ∥ g , C 2 = ⟨ f , f ⟩ g , C β \beta β b b b ∥ r b ∥ 2 ≤ C h β \lVert r_b\rVert_2\le Ch^\beta ∥ r b ∥ 2 ≤ C h β ∥ S ∥ o p ≤ g − − ( p a + p b + 1 ) \lVert S\rVert_{\mathrm{op}}\le g_-^{-(p_a+p_b+1)} ∥ S ∥ op ≤ g − − ( p a + p b + 1 ) ∣ F C ∣ ≤ C h α + β \lvert F_C\rvert\le Ch^{\alpha+\beta} ∣ F C ∣ ≤ C h α + β a a a S S S 1 / det Γ a 1/\det\Gamma_a 1/ det Γ a 1 / det Γ b 1/\det\Gamma_b 1/ det Γ b m + 1 m+1 m + 1 ν = q α + q β \nu=q_\alpha+q_\beta ν = q α + q β
Polynomial approximation. We apply Lemma 4.2 (a) with N = 2 N=2 N = 2 ( M 0 , M 1 , M 2 ) = ( Ω , Γ a , Γ b ) (M_0,M_1,M_2)=(\Omega,\Gamma_a,\Gamma_b) ( M 0 , M 1 , M 2 ) = ( Ω , Γ a , Γ b ) ν ′ = p a + p b \nu'=p_a+p_b ν ′ = p a + p b
F C , m ( m C ) = r a ⊤ S m ( Ω , L a ⊤ Ω L a , L b ⊤ Ω L b ) r b . (39) F_{C,m}(m_C)=r_a^\top S_m(\Omega,L_a^\top\Omega L_a,L_b^\top\Omega L_b)r_b. \tag*{(39)} F C , m ( m C ) = r a ⊤ S m ( Ω , L a ⊤ Ω L a , L b ⊤ Ω L b ) r b . (39) The substitutions Γ η = L η ⊤ Ω L η \Gamma_\eta=L_\eta^\top\Omega L_\eta Γ η = L η ⊤ Ω L η F C , m F_{C,m} F C , m m + p a + p b + 2 m+p_a+p_b+2 m + p a + p b + 2 F C − F C , m = r a ⊤ ( S − S m ) r b F_C-F_{C,m}=r_a^\top(S-S_m)r_b F C − F C , m = r a ⊤ ( S − S m ) r b r a , r b r_a,r_b r a , r b (30) imply that ∣ F C − F C , m ∣ ≤ C h α + β ( m + 1 ) p a + p b ρ m \lvert F_C-F_{C,m}\rvert\le Ch^{\alpha+\beta}(m+1)^{p_a+p_b}\rho^m ∣ F C − F C , m ∣ ≤ C h α + β ( m + 1 ) p a + p b ρ m p a + p b = ν p_a+p_b=\nu p a + p b = ν (33) ; for the base term, p a = p b = 0 p_a=p_b=0 p a = p b = 0 m + 2 m+2 m + 2 C ρ m C\rho^m C ρ m
The complex bound and the gradient. For each of the finitely many choices of ( L a , L b ) (L_a,L_b) ( L a , L b ) V = { t : g − I ⪯ Ω ⪯ g + I , ∥ u ∥ 2 , ∥ v ∥ 2 ≤ H g + } \mathcal{V}=\{t:g_-I\preceq\Omega\preceq g_+I,\lVert u\rVert_2,\lVert v\rVert_2\le Hg_+\} V = { t : g − I ⪯ Ω ⪯ g + I , ∥ u ∥ 2 , ∥ v ∥ 2 ≤ H g + } Ω , u , v \Omega,u,v Ω , u , v t t t m C m_C m C m C ( P ) m_C(\mathcal{P}) m C ( P ) P ∈ P ( α , β ) P\in\mathcal{P}(\alpha,\beta) P ∈ P ( α , β ) Ω \Omega Ω L a ⊤ Ω L a L_a^\top\Omega L_a L a ⊤ Ω L a L b ⊤ Ω L b L_b^\top\Omega L_b L b ⊤ Ω L b [ g − , g + ] [g_-,g_+] [ g − , g + ] (29) , F C , m F_{C,m} F C , m S m S_m S m m m m z z z
g ˉ − ( ν ′ + 1 ) r a ⊤ K ( z ; Ω ) r b det K ( z ; L a ⊤ Ω L a ) det K ( z ; L b ⊤ Ω L b ) \bar{g}^{-(\nu'+1)}r_a^\top\mathcal{K}(z;\Omega)r_b\det\mathcal{K}(z;L_a^\top\Omega L_a)\det\mathcal{K}(z;L_b^\top\Omega L_b) g ˉ − ( ν ′ + 1 ) r a ⊤ K ( z ; Ω ) r b det K ( z ; L a ⊤ Ω L a ) det K ( z ; L b ⊤ Ω L b ) and of the entries of the corresponding matrix, in which each determinant is a fixed sum of products of kernel entries, or one, and the entries of r a , r b r_a,r_b r a , r b t t t Lemma 4.2 (b), there exist ϵ ∈ ( 0 , 1 ] \epsilon\in(0,1] ϵ ∈ ( 0 , 1 ] C C C F C , m F_{C,m} F C , m S m S_m S m C C C ϵ \epsilon ϵ V \mathcal{V} V m m m (34) . By Lemma 4.2 (c) with r = 1 r=1 r = 1 S m S_m S m V \mathcal{V} V r a , r b r_a,r_b r a , r b m C ( P ) m_C(P) m C ( P )
∥ ∇ F C , m ∥ 2 ≤ C ( ∥ r a ∥ 2 + ∥ r b ∥ 2 + ∥ r a ∥ 2 ∥ r b ∥ 2 ) ≤ C h s 0 . \lVert\nabla F_{C,m}\rVert_2\le C\left(\lVert r_a\rVert_2+\lVert r_b\rVert_2+\lVert r_a\rVert_2\lVert r_b\rVert_2\right)\le Ch^{s_0}. ∥ ∇ F C , m ∥ 2 ≤ C ( ∥ r a ∥ 2 + ∥ r b ∥ 2 + ∥ r a ∥ 2 ∥ r b ∥ 2 ) ≤ C h s 0 . which is (35) . Let C 0 ≥ 1 C_0\ge1 C 0 ≥ 1
For 0 < α , β ≤ 1 0 < \alpha,\beta\le1 0 < α , β ≤ 1 g g g C 1 , C 2 C_1,C_2 C 1 , C 2 C C C x i = 2 K ∫ C i g x_i = 2K\int_{C_i} g x i = 2 K ∫ C i g u i = 2 K ∫ C i a g u_i = 2K\int_{C_i} ag u i = 2 K ∫ C i a g v i = 2 K ∫ C i b g v_i = 2K\int_{C_i} bg v i = 2 K ∫ C i b g x ˉ = ( x 1 + x 2 ) / 2 \bar{x} = (x_1+x_2)/2 x ˉ = ( x 1 + x 2 ) /2 z C = ( 2 1 C 1 , 2 1 C 2 ) T z_C = (\sqrt{2}\,1_{C_1},\sqrt{2}\,1_{C_2})^{\mathsf{T}} z C = ( 2 1 C 1 , 2 1 C 2 ) T L a = L b = ( 1 , 1 ) T / 2 L_a=L_b=(1,1)^{\mathsf{T}}/\sqrt{2} L a = L b = ( 1 , 1 ) T / 2 Ω = diag ( x 1 , x 2 ) \Omega=\operatorname{diag}(x_1,x_2) Ω = diag ( x 1 , x 2 ) u = ( u 1 , u 2 ) T / 2 u=(u_1,u_2)^{\mathsf{T}}/\sqrt{2} u = ( u 1 , u 2 ) T / 2 Γ a = Γ b = x ˉ \Gamma_a=\Gamma_b=\bar{x} Γ a = Γ b = x ˉ r a = A C ( 1 , − 1 ) T / ( 2 2 ) r_a=A_C(1,-1)^{\mathsf{T}}/(2\sqrt{2}) r a = A C ( 1 , − 1 ) T / ( 2 2 )
F C = A C B C 4 x 1 x 2 x ˉ , A C = x 2 u 1 − x 1 u 2 , B C = x 2 v 1 − x 1 v 2 . (40) F_C=\frac{A_CB_C}{4x_1x_2\bar{x}},\qquad A_C=x_2u_1-x_1u_2,\qquad B_C=x_2v_1-x_1v_2. \tag*{(40)} F C = 4 x 1 x 2 x ˉ A C B C , A C = x 2 u 1 − x 1 u 2 , B C = x 2 v 1 − x 1 v 2 . (40) The contrasts A C = x 1 x 2 ( u 1 / x 1 − u 2 / x 2 ) A_C=x_1x_2(u_1/x_1-u_2/x_2) A C = x 1 x 2 ( u 1 / x 1 − u 2 / x 2 ) B C B_C B C g g g a a a b b b
Unbiased estimation and variance
Let O 1 , … , O n O_1,\ldots,O_n O 1 , … , O n Z Z Z R p \mathbb{R}^p R p m = E P Z ( O ) m=\mathbb{E}_PZ(O) m = E P Z ( O ) F F F R p \mathbb{R}^p R p R R R F ( m ) = ∑ q = 0 R D q F ( 0 ) [ m , … , m ] / q ! F(m)=\sum_{q=0}^{R}D^qF(0)[m,\ldots,m]/q! F ( m ) = ∑ q = 0 R D q F ( 0 ) [ m , … , m ] / q ! 0 ≤ q ≤ n 0\le q\le n 0 ≤ q ≤ n ( n ) q = n ( n − 1 ) ⋯ ( n − q + 1 ) (n)_q=n(n-1)\cdots(n-q+1) ( n ) q = n ( n − 1 ) ⋯ ( n − q + 1 ) ( n ) 0 = 1 (n)_0=1 ( n ) 0 = 1 R ≤ n R\le n R ≤ n F F F
F ^ = ∑ q = 0 R 1 q ! ( n ) q ∑ i 1 , … , i q ≤ n ≠ D q F ( 0 ) [ Z ( O i 1 ) , … , Z ( O i q ) ] , (41) \widehat{F}=\sum_{q=0}^{R}\frac{1}{q!(n)_q}\sum_{i_1,\ldots,i_q\le n}^{\ne}D^qF(0)[Z(O_{i_1}),\ldots,Z(O_{i_q})], \tag*{(41)} F = q = 0 ∑ R q ! ( n ) q 1 i 1 , … , i q ≤ n ∑ = D q F ( 0 ) [ Z ( O i 1 ) , … , Z ( O i q )] , (41) where ∑ ≠ \sum^{\ne} ∑ = q q q { 1 , … , n } \{1,\ldots,n\} { 1 , … , n } q = 0 q=0 q = 0 F ( 0 ) F(0) F ( 0 ) [17 , 19 ] , is linear in F F F Z Z Z
Lemma 4.4 (Mean and covariance of the lift). Let F 1 , F 2 F_1,F_2 F 1 , F 2 R p \mathbb{R}^p R p n n n Y l = Z ( O l ′ ) − m Y_l=Z(O'_l)-m Y l = Z ( O l ′ ) − m O 1 ′ , O 2 ′ , … O'_1,O'_2,\ldots O 1 ′ , O 2 ′ , … P P P E F ^ 1 = F 1 ( m ) \mathbb{E}\widehat{F}_1=F_1(m) E F 1 = F 1 ( m )
Cov ( F ^ 1 , F ^ 2 ) = ∑ r ≥ 1 1 r ! ( n ) r E [ D r F 1 ( m ) [ Y 1 , … , Y r ] D r F 2 ( m ) [ Y 1 , … , Y r ] ] , \operatorname{Cov}(\widehat{F}_1,\widehat{F}_2)=\sum_{r\ge1}\frac{1}{r!(n)_r}\mathbb{E}\left[D^rF_1(m)[Y_1,\ldots,Y_r]D^rF_2(m)[Y_1,\ldots,Y_r]\right], Cov ( F 1 , F 2 ) = r ≥ 1 ∑ r ! ( n ) r 1 E [ D r F 1 ( m ) [ Y 1 , … , Y r ] D r F 2 ( m ) [ Y 1 , … , Y r ] ] , where the sum stops at the smaller degree.
Proof. Let Y i = Z ( O i ) − m Y_i=Z(O_i)-m Y i = Z ( O i ) − m F F F F 1 F_1 F 1 F 2 F_2 F 2 Z ( O i ) = m + Y i Z(O_i)=m+Y_i Z ( O i ) = m + Y i r r r ( k 1 , … , k r ) (k_1,\ldots,k_r) ( k 1 , … , k r ) ( q r ) ( n − r ) q − r \binom{q}{r}(n-r)_{q-r} ( r q ) ( n − r ) q − r q q q D q F ( 0 ) D^qF(0) D q F ( 0 ) D q F ( 0 ) [ Y k 1 , … , Y k r , m , … , m ] D^qF(0)[Y_{k_1},\ldots,Y_{k_r},m,\ldots,m] D q F ( 0 ) [ Y k 1 , … , Y k r , m , … , m ] ( q r ) ( n − r ) q − r / { q ! ( n ) q } = 1 / { r ! ( q − r ) ! ( n ) r } \binom{q}{r}(n-r)_{q-r}/\{q!(n)_q\}=1/\{r!(q-r)!(n)_r\} ( r q ) ( n − r ) q − r / { q ! ( n ) q } = 1/ { r ! ( q − r )! ( n ) r } D r F ( m ) [ y 1 , … , y r ] = ∑ q ≥ r D q F ( 0 ) [ y 1 , … , y r , m , … , m ] / ( q − r ) ! D^rF(m)[y_1,\ldots,y_r]=\sum_{q\ge r}D^qF(0)[y_1,\ldots,y_r,m,\ldots,m]/(q-r)! D r F ( m ) [ y 1 , … , y r ] = ∑ q ≥ r D q F ( 0 ) [ y 1 , … , y r , m , … , m ] / ( q − r )!
F ^ = F ( m ) + ∑ r = 1 deg F 1 r ! ( n ) r ∑ k 1 , … , k r ≤ n ≠ D r F ( m ) [ Y k 1 , … , Y k r ] . \widehat{F}=F(m)+\sum_{r=1}^{\deg F}\frac{1}{r!(n)_r}\sum_{k_1,\ldots,k_r\le n}^{\ne}D^rF(m)[Y_{k_1},\ldots,Y_{k_r}]. F = F ( m ) + r = 1 ∑ d e g F r ! ( n ) r 1 k 1 , … , k r ≤ n ∑ = D r F ( m ) [ Y k 1 , … , Y k r ] . The Y i Y_i Y i r ≥ 1 r \ge1 r ≥ 1 E F ^ 1 = F 1 ( m ) \mathbb{E}\widehat{F}_1 = F_1(\mathbf{m}) E F 1 = F 1 ( m ) r r r ( n r ) r ! \binom{n}{r}r! ( r n ) r ! E [ D r F 1 ( m ) [ Y 1 , … , Y r ] D r F 2 ( m ) [ Y 1 , … , Y r ] ] / { r ! ( n r ) } 2 \mathbb{E}[D^rF_1(\mathbf{m})[Y_1,\ldots,Y_r]D^rF_2(\mathbf{m})[Y_1,\ldots,Y_r]]/\{r!\binom{n}{r}\}^2 E [ D r F 1 ( m ) [ Y 1 , … , Y r ] D r F 2 ( m ) [ Y 1 , … , Y r ]] / { r ! ( r n ) } 2
The expansion in this proof is the Hoeffding decomposition of the U-statistics in (4.16) into degenerate U-statistics; see [67 Sections 11.4 and 12.3] and [58 Section 5.1.5, Lemma A] . Hoeffding [19 Sections 5–6] derived exact variance and covariance formulas by grouping pairs of index sets according to their overlap. A covariance identity of the same form as that of Lemma 4.4, for linear combinations of complete U-statistics estimating trace polynomials, appears in [60 Theorem 6.2] .
For a multilinear form H H H H [ Y 1 , … , Y r ] H[Y_1,\ldots,Y_r] H [ Y 1 , … , Y r ] H [ Z ( O 1 ′ ) , … , Z ( O r ′ ) ] H[Z(O'_1),\ldots,Z(O'_r)] H [ Z ( O 1 ′ ) , … , Z ( O r ′ )] ∏ l = 1 r ( I − E l ) \prod_{l=1}^r(I-\mathcal{E}_l) ∏ l = 1 r ( I − E l ) E l \mathcal{E}_l E l l l l P P P L 2 ( P ⊗ r ) L^2(P^{\otimes r}) L 2 ( P ⊗ r )
E ∣ H [ Y 1 , … , Y r ] ∣ 2 ≤ E ∣ H [ Z ( O 1 ′ ) , … , Z ( O r ′ ) ] ∣ 2 . (42) \mathbb{E}|H[Y_1,\ldots,Y_r]|^2 \le\mathbb{E}|H[Z(O'_1),\ldots,Z(O'_r)]|^2. \tag*{(42)} E ∣ H [ Y 1 , … , Y r ] ∣ 2 ≤ E ∣ H [ Z ( O 1 ′ ) , … , Z ( O r ′ )] ∣ 2 . (42) This allows us to bound the variance using the cell support of Z Z Z Z C ( O ) − m C Z_C(O)-m_C Z C ( O ) − m C C C C H [ Y 1 , … , Y r ] H[Y_1,\ldots,Y_r] H [ Y 1 , … , Y r ] H [ Z ( O 1 ′ ) , … , Z ( O r ′ ) ] H[Z(O'_1),\ldots,Z(O'_r)] H [ Z ( O 1 ′ ) , … , Z ( O r ′ )] [67 Section 11.4] , and (4.17) is the corresponding contraction property of that decomposition.
Lemma 4.5 (Variance of a polynomial in cell moments). Fix integers n n n K K K R ≥ 1 R \ge1 R ≥ 1 2 R ≤ n 2R \le n 2 R ≤ n h ∈ ( 0 , 1 ] h \in(0,1] h ∈ ( 0 , 1 ] [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d K K K C C C Z C Z_C Z C m C = E P Z C ( O ) m_C=\mathbb{E}_PZ_C(O) m C = E P Z C ( O ) F C F_C F C R R R T ^ \widehat{T} T T = K − 1 ∑ C F C ( m C ) T=K^{-1}\sum_C F_C(m_C) T = K − 1 ∑ C F C ( m C ) ( m C ) C (m_C)_C ( m C ) C Z = ( Z C ) C Z=(Z_C)_C Z = ( Z C ) C C 0 ≥ 1 C_0 \ge1 C 0 ≥ 1 s 0 > 0 s_0>0 s 0 > 0
∥ Z C ( O ) ∥ 2 ≤ C 0 K 1 C ( X ) , P ( X ∈ C ) ≤ C 0 / K , \lVert Z_C(O)\rVert_2 \le C_0K\mathbf{1}_C(X), \qquad P(X \in C) \le C_0/K, ∥ Z C ( O ) ∥ 2 ≤ C 0 K 1 C ( X ) , P ( X ∈ C ) ≤ C 0 / K , that ∣ F C ( t ) ∣ ≤ C 0 |F_C(t)| \le C_0 ∣ F C ( t ) ∣ ≤ C 0 t t t ∥ t − m C ∥ ∞ < 1 / C 0 \lVert t-m_C\rVert_\infty< 1/C_0 ∥ t − m C ∥ ∞ < 1/ C 0 ∥ ∇ F C ( m C ) ∥ 2 ≤ C 0 h s 0 \lVert\nabla F_C(m_C)\rVert_2 \le C_0h^{s_0} ∥ ∇ F C ( m C ) ∥ 2 ≤ C 0 h s 0 C v = 2 e 2 C 0 7 C_v=2e^2C_0^7 C v = 2 e 2 C 0 7
Var T ^ ≤ C v h 2 s 0 n + 1 n ∑ r = 2 R C v r r ! ( K n ) r − 1 . (43) \operatorname{Var}\widehat{T} \le\frac{C_vh^{2s_0}}{n}+\frac{1}{n}\sum_{r=2}^{R}C_v^r r!\left(\frac{K}{n}\right)^{r-1}. \tag*{(43)} Var T ≤ n C v h 2 s 0 + n 1 r = 2 ∑ R C v r r ! ( n K ) r − 1 . (43) If moreover C v R K / n ≤ 1 / 2 C_vRK/n \le1/2 C v R K / n ≤ 1/2
Var T ^ ≤ C v h 2 s 0 n + 4 C v 2 K n 2 . (44) \operatorname{Var}\widehat{T} \le\frac{C_vh^{2s_0}}{n}+\frac{4C_v^2K}{n^2}. \tag*{(44)} Var T ≤ n C v h 2 s 0 + n 2 4 C v 2 K . (44) Proof. We write m = ( m C ) C \mathbf{m}=(m_C)_C m = ( m C ) C H r = D r T ( m ) [ Z ( O 1 ′ ) , … , Z ( O r ′ ) ] H_r=D^rT(\mathbf{m})[Z(O'_1),\ldots,Z(O'_r)] H r = D r T ( m ) [ Z ( O 1 ′ ) , … , Z ( O r ′ )] (42) , the term of order r r r T ^ \widehat{T} T E ∣ H r ∣ 2 / { r ! ( n ) r } \mathbb{E}|H_r|^2/\{r!(n)_r\} E ∣ H r ∣ 2 / { r ! ( n ) r } r > R r>R r > R o l = ( x l , ζ l ) o_l=(x_l,\zeta_l) o l = ( x l , ζ l )
D r T ( m ) [ Z ( o 1 ) , … , Z ( o r ) ] = K − 1 ∑ C D r F C ( m C ) [ Z C ( o 1 ) , … , Z C ( o r ) ] . D^rT(\mathbf{m})[Z(o_1),\ldots,Z(o_r)]=K^{-1}\sum_C D^rF_C(m_C)[Z_C(o_1),\ldots,Z_C(o_r)]. D r T ( m ) [ Z ( o 1 ) , … , Z ( o r )] = K − 1 C ∑ D r F C ( m C ) [ Z C ( o 1 ) , … , Z C ( o r )] . Since the cells are disjoint and Z C Z_C Z C C C C C C C x 1 , … , x r ∈ C x_1,\ldots,x_r\in C x 1 , … , x r ∈ C
∣ D r T ( m ) [ Z ( o 1 ) , … , Z ( o r ) ] ∣ 2 = K − 2 ∑ C ∣ D r F C ( m C ) [ Z C ( o 1 ) , … , Z C ( o r ) ] ∣ 2 . \left|D^rT(\mathbf{m})[Z(o_1),\ldots,Z(o_r)]\right|^2=K^{-2}\sum_C\left|D^rF_C(m_C)[Z_C(o_1),\ldots,Z_C(o_r)]\right|^2. ∣ D r T ( m ) [ Z ( o 1 ) , … , Z ( o r )] ∣ 2 = K − 2 C ∑ ∣ D r F C ( m C ) [ Z C ( o 1 ) , … , Z C ( o r )] ∣ 2 . For r ≥ 2 r\ge2 r ≥ 2 ε = 1 / C 0 \varepsilon=1/C_0 ε = 1/ C 0 A = C 0 A=C_0 A = C 0 ∥ Z C ∥ ∞ ≤ ∥ Z C ∥ 2 ≤ C 0 K \|Z_C\|_\infty\le\|Z_C\|_2\le C_0K ∥ Z C ∥ ∞ ≤ ∥ Z C ∥ 2 ≤ C 0 K C C C C 0 2 ( r ! ) 2 ( e C 0 ) 2 r ( C 0 K ) 2 r C_0^2(r!)^2(eC_0)^{2r}(C_0K)^{2r} C 0 2 ( r ! ) 2 ( e C 0 ) 2 r ( C 0 K ) 2 r X 1 ′ , … , X r ′ ∈ C X'_1,\ldots,X'_r\in C X 1 ′ , … , X r ′ ∈ C ( C 0 / K ) r (C_0/K)^r ( C 0 / K ) r K K K
E ∣ H r ∣ 2 ≤ C 0 2 ( r ! ) 2 ( e 2 C 0 5 ) r K r − 1 ≤ ( r ! ) 2 ( e 2 C 0 7 ) r K r − 1 . \mathbb{E}|H_r|^2\le C_0^2(r!)^2(e^2C_0^5)^rK^{r-1}\le(r!)^2(e^2C_0^7)^rK^{r-1}. E ∣ H r ∣ 2 ≤ C 0 2 ( r ! ) 2 ( e 2 C 0 5 ) r K r − 1 ≤ ( r ! ) 2 ( e 2 C 0 7 ) r K r − 1 . For r = 1 r=1 r = 1 ∣ ∇ F C ( m C ) ⋅ Z C ( o ) ∣ 2 ≤ C 0 4 h 2 s 0 K 2 1 C ( x ) \left|\nabla F_C(m_C)\cdot Z_C(o)\right|^2\le C_0^4h^{2s_0}K^2 1_C(x) ∣ ∇ F C ( m C ) ⋅ Z C ( o ) ∣ 2 ≤ C 0 4 h 2 s 0 K 2 1 C ( x ) [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d E ∣ H 1 ∣ 2 ≤ C 0 4 h 2 s 0 ≤ C ν h 2 s 0 \mathbb{E}|H_1|^2\le C_0^4h^{2s_0}\le C_\nu h^{2s_0} E ∣ H 1 ∣ 2 ≤ C 0 4 h 2 s 0 ≤ C ν h 2 s 0 2 R ≤ n 2R\le n 2 R ≤ n ( n ) r ≥ ( n / 2 ) r (n)_r\ge(n/2)^r ( n ) r ≥ ( n /2 ) r r ≤ R r\le R r ≤ R (43) . In (43) , the ratio of the ( r + 1 ) (r+1) ( r + 1 ) r r r C ν ( r + 1 ) K / n ≤ C ν R K / n C_\nu(r+1)K/n\le C_\nu RK/n C ν ( r + 1 ) K / n ≤ C ν R K / n r + 1 ≤ R r+1\le R r + 1 ≤ R 1 / 2 1/2 1/2 r ≥ 2 r\ge2 r ≥ 2 2 C ν 2 K / n 2 2C_\nu^2K/n^2 2 C ν 2 K / n 2 (44) . □
For a fixed order r r r n − 1 max { 1 , ( K / n ) r − 1 } n^{-1}\max\{1,(K/n)^{r-1}\} n − 1 max { 1 , ( K / n ) r − 1 } K K K [48 Theorems 3.20–3.21] . In (43) , we keep the dependence on the order explicit, including the factorial, since our orders grow with n n n
Choice of the degrees
For a level j ≥ 1 j\ge1 j ≥ 1 m ≥ 0 m\ge0 m ≥ 0 K j = 2 j − 1 K_j=2^{j-1} K j = 2 j − 1 h j = K j − 1 / d h_j=K_j^{-1/d} h j = K j − 1/ d T j , m ( P ) = K j − 1 ∑ C ∈ C j − 1 F C , m ( m C ) T_{j,m}(P)=K_j^{-1}\sum_{C\in\mathcal{C}_{j-1}}F_{C,m}(m_C) T j , m ( P ) = K j − 1 ∑ C ∈ C j − 1 F C , m ( m C ) m m m ψ j − ψ j − 1 \psi_j-\psi_{j-1} ψ j − ψ j − 1 K 0 = h 0 = 1 K_0=h_0=1 K 0 = h 0 = 1 T 0 , m ( P ) = F 0 , m ( m 0 ) T_{0,m}(P)=F_{0,m}(m_0) T 0 , m ( P ) = F 0 , m ( m 0 ) K 0 = K 1 = 1 K_0=K_1=1 K 0 = K 1 = 1 T ^ j , m \widehat{T}_{j,m} T j , m T j , m T_{j,m} T j , m C ν = 2 e 2 C 0 C_\nu=2e^2C_0 C ν = 2 e 2 C 0 C 0 C_0 C 0 R = m + ν + 2 R=m+\nu+2 R = m + ν + 2 2 R ≤ n 2R\le n 2 R ≤ n T j , m T_{j,m} T j , m K = K j K=K_j K = K j h = h j h=h_j h = h j R R R r ! ≤ R r − 1 r!\le R^{r-1} r ! ≤ R r − 1 2 ≤ r ≤ R 2\le r\le R 2 ≤ r ≤ R (43) then implies
Var T ^ j , m ≤ C ν n ( h j 2 s 0 + ∑ r = 1 R − 1 y r ) , y = C ν R K j n . (45) \operatorname{Var}\widehat{T}_{j,m}\le\frac{C_\nu}{n}\left(h_j^{2s_0}+\sum_{r=1}^{R-1}y^r\right),\qquad y=\frac{C_\nu RK_j}{n}. \tag*{(45)} Var T j , m ≤ n C ν ( h j 2 s 0 + r = 1 ∑ R − 1 y r ) , y = n C ν R K j . (45) For a terminal level J J J m 0 , … , m J m_0,\ldots,m_J m 0 , … , m J m j + ν + 2 ≤ n / 2 m_j+\nu+2\le n/2 m j + ν + 2 ≤ n /2 T ( P ) = ∫ a b g T(P)=\int abg T ( P ) = ∫ ab g T ^ = ∑ j = 0 J T ^ j , m j \widehat{T}=\sum_{j=0}^{J}\widehat{T}_{j,m_j} T = ∑ j = 0 J T j , m j ∑ j = 0 J T j , m j \sum_{j=0}^{J}T_{j,m_j} ∑ j = 0 J T j , m j ∑ j = 0 J T j , m j ( P ) \sum_{j=0}^{J}T_{j,m_j}(P) ∑ j = 0 J T j , m j ( P ) ψ J \psi_J ψ J (31) , (33) , the base-term bound, and h j α + β = K j − θ h_j^{\alpha+\beta}=K_j^{-\theta} h j α + β = K j − θ
∣ E T ^ − T ( P ) ∣ ≤ C 2 − J θ + C 0 ∑ j = 0 J f j ( m j ) , f j ( m ) = K j − θ ( m + 1 ) ν ρ m . (46) \left|\mathbb{E}\widehat{T}-T(P)\right|\le C2^{-J\theta}+C_0\sum_{j=0}^{J}f_j(m_j),\qquad f_j(m)=K_j^{-\theta}(m+1)^\nu\rho^m. \tag*{(46)} E T − T ( P ) ≤ C 2 − J θ + C 0 j = 0 ∑ J f j ( m j ) , f j ( m ) = K j − θ ( m + 1 ) ν ρ m . (46) By the L 2 L^2 L 2 T ^ \widehat{T} T T ^ j , m j \widehat{T}_{j,m_j} T j , m j
We choose the order on each level as the smaller of two values. The first, a bias rule, makes the approximation errors decrease geometrically from the terminal level toward the coarse levels. The second, a variance rule, is needed only below the threshold. We keep high orders only at coarse resolutions, and in this respect our estimator resembles that of Robins et al. [52 Section 9, (9.6)] . Their kernel of each order three or higher is a sum over the contiguous pairs of projection kernels in its product; in each summand, that pair is truncated below a hyperbola in the indices of the basis functions, and all other kernels are truncated at n n n
On a level with K K K R R R (45) is at most about n − 2 θ n^{-2\theta} n − 2 θ y R ≤ n 1 − 2 θ y^R\le n^{1-2\theta} y R ≤ n 1 − 2 θ x = log y > 0 x=\log y>0 x = log y > 0 ( 1 − 2 θ ) log n / x (1-2\theta)\log n/x ( 1 − 2 θ ) log n / x K = n ε x / ( C v R ) K=n^{\varepsilon x}/(C_vR) K = n ε x / ( C v R ) K − θ ρ m K^{-\theta}\rho^m K − θ ρ m n − θ ( C v R ) θ exp { − θ x − τ ( 1 − 2 θ ) log n / x } n^{-\theta}(C_vR)^\theta\exp\{-\theta x-\tau(1-2\theta)\log n/x\} n − θ ( C v R ) θ exp { − θ x − τ ( 1 − 2 θ ) log n / x } θ x + τ ( 1 − 2 θ ) log n / x \theta x+\tau(1-2\theta)\log n/x θ x + τ ( 1 − 2 θ ) log n / x x > 0 x>0 x > 0 κ log n \kappa\sqrt{\log n} κ log n R R R log n \sqrt{\log n} log n ( log n ) θ / 2 + ν / 2 + 1 / 4 (\log n)^{\theta/2+\nu/2+1/4} ( log n ) θ /2 + ν /2 + 1/4 a n ( log n ) ν / 2 + 1 / 4 / ( n − θ e − κ log n ) a_n(\log n)^{\nu/2+1/4}/(n^{-\theta}e^{-\kappa\sqrt{\log n}}) a n ( log n ) ν /2 + 1/4 / ( n − θ e − κ l o g n ) ( C v R ) θ (C_vR)^\theta ( C v R ) θ ( m + 1 ) ν (m+1)^\nu ( m + 1 ) ν (33) , and from the sum over levels. The next lemma bounds this sum. It is the only source of the factor ( log n ) 1 / 4 (\log n)^{1/4} ( log n ) 1/4
Lemma 4.6 (Window sum). Let θ , τ , B , X ˉ > 0 \theta,\tau,B,\bar X>0 θ , τ , B , X ˉ > 0 S ⊂ ( 0 , X ˉ ] S\subset(0,\bar X] S ⊂ ( 0 , X ˉ ] log 2 \log2 log 2
∑ x ∈ S exp ( − θ x − τ B x ) ≤ ( 2 + ( π X ˉ / θ ) 1 / 2 log 2 ) e − 2 θ τ B . \sum_{x\in S}\exp\left(-\theta x-\frac{\tau B}{x}\right)\le\left(2+\frac{(\pi\bar X/\theta)^{1/2}}{\log2}\right)e^{-2\sqrt{\theta\tau B}}. x ∈ S ∑ exp ( − θ x − x τ B ) ≤ ( 2 + log 2 ( π X ˉ / θ ) 1/2 ) e − 2 θ τ B . Proof. Let ξ = τ B / θ \xi=\sqrt{\tau B/\theta} ξ = τ B / θ
θ x + τ B x = 2 θ τ B + θ ( x − ξ ) 2 x ≥ 2 θ τ B + θ ( x − ξ ) 2 X ˉ , 0 < x ≤ X ˉ . \theta x+\frac{\tau B}{x}=2\sqrt{\theta\tau B}+\frac{\theta(x-\xi)^2}{x}\ge2\sqrt{\theta\tau B}+\frac{\theta(x-\xi)^2}{\bar X},\qquad0<x\le\bar X. θ x + x τ B = 2 θ τ B + x θ ( x − ξ ) 2 ≥ 2 θ τ B + X ˉ θ ( x − ξ ) 2 , 0 < x ≤ X ˉ . We list the points of S S S [ ξ , ∞ ) [\xi,\infty) [ ξ , ∞ ) ( 0 , ξ ) (0,\xi) ( 0 , ξ ) k k k k = 0 k=0 k = 0 ∣ x − ξ ∣ ≥ k log 2 |x-\xi|\ge k\log2 ∣ x − ξ ∣ ≥ k log 2 c = θ ( log 2 ) 2 / X ˉ c=\theta(\log2)^2/\bar X c = θ ( log 2 ) 2 / X ˉ 2 e − 2 θ τ B ∑ k ≥ 0 e − c k 2 2e^{-2\sqrt{\theta\tau B}}\sum_{k\ge0}e^{-ck^2} 2 e − 2 θ τ B ∑ k ≥ 0 e − c k 2 ∑ k ≥ 0 e − c k 2 ≤ 1 + ∫ 0 ∞ e − c u 2 d u = 1 + 1 2 ( π / c ) 1 / 2 \sum_{k\ge0}e^{-ck^2}\le1+\int_0^\infty e^{-cu^2}\,du=1+\frac{1}{2}(\pi/c)^{1/2} ∑ k ≥ 0 e − c k 2 ≤ 1 + ∫ 0 ∞ e − c u 2 d u = 1 + 2 1 ( π / c ) 1/2
The construction below also gives estimators satisfying, for every integer n ≥ 3 n \ge3 n ≥ 3
sup P ∈ P ( α , β ) ∥ ψ ^ n − ψ ( P ) ∥ L 2 ( P n ) ≤ C { n − 1 / 2 + n − θ } . (47) \sup_{P \in\mathcal{P}(\alpha,\beta)} \|\widehat{\psi}_n-\psi(P)\|_{L^2(P^n)} \le C\{n^{-1/2}+n^{-\theta}\}. \tag*{(47)} P ∈ P ( α , β ) sup ∥ ψ n − ψ ( P ) ∥ L 2 ( P n ) ≤ C { n − 1/2 + n − θ } . (47) Moreover, fix 0 ≤ M W < ∞ 0 \le M_W < \infty 0 ≤ M W < ∞ G ⊂ { ( x , y ) : 0 < x < y < ∞ } \mathcal{G} \subset\{(x,y):0 < x < y < \infty\} G ⊂ {( x , y ) : 0 < x < y < ∞ } C C C ( g − , g + ) ∈ G (g_-,g_+) \in\mathcal{G} ( g − , g + ) ∈ G ∥ W ∥ ∞ ≤ M W \|W\|_\infty\le M_W ∥ W ∥ ∞ ≤ M W ρ \rho ρ τ \tau τ κ \kappa κ a n a_n a n X X X
Proof of Theorem 2.3 (a). We write L = log n L=\log n L = log n r ˉ n = n − 1 / 2 \bar r_n=n^{-1/2} r ˉ n = n − 1/2 θ ≥ 1 / 2 \theta\ge1/2 θ ≥ 1/2 r ˉ n = a n L ν / 2 + 1 / 4 = n − θ L θ / 2 + ν / 2 + 1 / 4 e − κ L \bar r_n=a_nL^{\nu/2+1/4}=n^{-\theta}L^{\theta/2+\nu/2+1/4}e^{-\kappa\sqrt{L}} r ˉ n = a n L ν /2 + 1/4 = n − θ L θ /2 + ν /2 + 1/4 e − κ L 0 < θ < 1 / 2 0<\theta<1/2 0 < θ < 1/2 n − 1 / 2 ≤ C r ˉ n n^{-1/2}\le C\bar r_n n − 1/2 ≤ C r ˉ n n − 1 / 2 / r ˉ n ≤ e − ( 1 / 2 − θ ) L + κ L n^{-1/2}/\bar r_n\le e^{-(1/2-\theta)L+\kappa\sqrt{L}} n − 1/2 / r ˉ n ≤ e − ( 1/2 − θ ) L + κ L T ^ \widehat T T T ( P ) = ∫ a b g T(P)=\int abg T ( P ) = ∫ ab g E ( T ^ − T ( P ) ) 2 ≤ C r ˉ n 2 \mathbb{E}(\widehat T-T(P))^2\le C\bar r_n^2 E ( T − T ( P ) ) 2 ≤ C r ˉ n 2 P ∈ P ( α , β ) P\in\mathcal{P}(\alpha,\beta) P ∈ P ( α , β ) n 1 ≥ 3 n_1\ge3 n 1 ≥ 3 n ≥ n 1 n\ge n_1 n ≥ n 1 3 ≤ n < n 1 3\le n<n_1 3 ≤ n < n 1 0 0 0 ∣ T ( P ) ∣ ≤ H 2 g + |T(P)|\le H^2g_+ ∣ T ( P ) ∣ ≤ H 2 g + r ˉ n \bar r_n r ˉ n n ≥ n 1 n\ge n_1 n ≥ n 1
The two rules. We fix A > 0 A>0 A > 0 τ A ≥ θ log 2 + 1 \tau A\ge\theta\log2+1 τ A ≥ θ log 2 + 1 K ∗ = 2 J K_* = 2^J K ∗ = 2 J l j = J − j + 1 l_j=J-j+1 l j = J − j + 1 0 ≤ j ≤ J 0\le j\le J 0 ≤ j ≤ J K j = K ∗ 2 − l j K_j=K_*2^{-l_j} K j = K ∗ 2 − l j j ≥ 1 j\ge1 j ≥ 1 K 0 = 1 = 2 K ∗ 2 − l 0 K_0=1=2K_*2^{-l_0} K 0 = 1 = 2 K ∗ 2 − l 0 K ∗ 2 − l j ≤ K j ≤ 2 K ∗ 2 − l j K_*2^{-l_j}\le K_j\le2K_*2^{-l_j} K ∗ 2 − l j ≤ K j ≤ 2 K ∗ 2 − l j j j j bias rule is the order ⌈ A l j ⌉ \lceil Al_j\rceil ⌈ A l j ⌉ ( ⌈ A l j ⌉ + 1 ) ν ≤ ( A l j + 2 ) ν (\lceil Al_j\rceil+1)^\nu\le(Al_j+2)^\nu (⌈ A l j ⌉ + 1 ) ν ≤ ( A l j + 2 ) ν ρ ⌈ A l j ⌉ ≤ e − τ A l j \rho^{\lceil Al_j\rceil}\le e^{-\tau Al_j} ρ ⌈ A l j ⌉ ≤ e − τ A l j
∑ j = 0 J f j ( ⌈ A l j ⌉ ) ≤ K ∗ − θ ∑ l ≥ 1 ( A l + 2 ) ν e − l = C A K ∗ − θ , (48) \sum_{j=0}^{J} f_j(\lceil Al_j\rceil)\le K_*^{-\theta}\sum_{l\ge1}(Al+2)^\nu e^{-l}=C_AK_*^{-\theta}, \tag*{(48)} j = 0 ∑ J f j (⌈ A l j ⌉) ≤ K ∗ − θ l ≥ 1 ∑ ( A l + 2 ) ν e − l = C A K ∗ − θ , (48) because the l j l_j l j τ A − θ log 2 ≥ 1 \tau A-\theta\log2\ge1 τ A − θ log 2 ≥ 1 j j j m j = min { ⌈ A l j ⌉ , M j } m_j=\min\{\lceil Al_j\rceil,M_j\} m j = min {⌈ A l j ⌉ , M j } variance rule M j ∈ { 0 , 1 , 2 , … } ∪ { ∞ } M_j\in\{0,1,2,\ldots\}\cup\{\infty\} M j ∈ { 0 , 1 , 2 , … } ∪ { ∞ } j j j capped if M j < ⌈ A l j ⌉ M_j<\lceil Al_j\rceil M j < ⌈ A l j ⌉ (46) , (48) , and 2 − J θ = K ∗ − θ 2^{-J\theta}=K_*^{-\theta} 2 − J θ = K ∗ − θ
∣ E T ^ − T ( P ) ∣ ≤ C K ∗ − θ + C ∑ j c a p p e d f j ( M j ) . (49) |\mathbb{E}\widehat T-T(P)|\le CK_*^{-\theta}+C\sum_{j\ \mathrm{capped}}f_j(M_j). \tag*{(49)} ∣ E T − T ( P ) ∣ ≤ C K ∗ − θ + C j capped ∑ f j ( M j ) . (49) The degree R j = m j + ν + 2 R_j=m_j+\nu+2 R j = m j + ν + 2 R j ≤ A l j + ν + 3 R_j\le Al_j+\nu+3 R j ≤ A l j + ν + 3 J ≤ C L J\le CL J ≤ C L R j ≤ C L ≤ n / 2 R_j\le CL\le n/2 R j ≤ C L ≤ n /2 (45) applies to level j j j y = y j = C v R j K j / n y=y_j=C_vR_jK_j/n y = y j = C v R j K j / n
The case θ ≥ 1 / 2 \theta\ge1/2 θ ≥ 1/2 Let c K ∈ ( 0 , 1 ] c_K\in(0,1] c K ∈ ( 0 , 1 ] c K ≤ { 4 C v sup l ≥ 1 ( A l + ν + 3 ) 2 2 − l } − 1 c_K\le\{4C_v\sup_{l\ge1}(Al+\nu+3)^2 2^{-l}\}^{-1} c K ≤ { 4 C v sup l ≥ 1 ( A l + ν + 3 ) 2 2 − l } − 1 J = ⌊ log 2 ( c K n ) ⌋ ≥ 0 J=\lfloor\log_2(c_Kn)\rfloor\ge0 J = ⌊ log 2 ( c K n )⌋ ≥ 0 M j = ∞ M_j=\infty M j = ∞ j j j K ∗ ≤ c K n K_*\le c_Kn K ∗ ≤ c K n C v R j K j / n ≤ 2 C v c K ( A l j + ν + 3 ) 2 − l j ≤ 1 / 2 C_vR_jK_j/n\le2C_vc_K(Al_j+\nu+3)2^{-l_j}\le1/2 C v R j K j / n ≤ 2 C v c K ( A l j + ν + 3 ) 2 − l j ≤ 1/2 (44) gives Var T ^ j , m j ≤ C v h j 2 s 0 / n + 4 C v 2 K j / n 2 \operatorname{Var}\widehat T_{j,m_j}\le C_vh_j^{2s_0}/n+4C_v^2K_j/n^2 Var T j , m j ≤ C v h j 2 s 0 / n + 4 C v 2 K j / n 2 ∑ j ≥ 0 h j s 0 ≤ 1 + ∑ i ≥ 0 2 − i s 0 / d \sum_{j\ge0}h_j^{s_0}\le1+\sum_{i\ge0}2^{-is_0/d} ∑ j ≥ 0 h j s 0 ≤ 1 + ∑ i ≥ 0 2 − i s 0 / d ∑ j = 0 J K j 1 / 2 ≤ C K ∗ 1 / 2 ≤ C n 1 / 2 \sum_{j=0}^{J}K_j^{1/2}\le CK_*^{1/2}\le Cn^{1/2} ∑ j = 0 J K j 1/2 ≤ C K ∗ 1/2 ≤ C n 1/2 T ^ \widehat T T C n − 1 / 2 Cn^{-1/2} C n − 1/2 (49) implies that the bias is at most C K ∗ − θ ≤ C ( c K n / 2 ) − θ ≤ C n − 1 / 2 CK_*^{-\theta}\le C(c_Kn/2)^{-\theta}\le Cn^{-1/2} C K ∗ − θ ≤ C ( c K n /2 ) − θ ≤ C n − 1/2
The case 0 < θ < 1 / 2 0<\theta<1/2 0 < θ < 1/2 Δ = 1 − 2 θ \Delta=1-2\theta Δ = 1 − 2 θ κ = 2 θ Δ τ \kappa=2\sqrt{\theta\Delta\tau} κ = 2 θ Δ τ
J = ⌈ log 2 n + κ L θ log 2 ⌉ , J=\left\lceil\log_{2}n+\frac{\kappa\sqrt{L}}{\theta\log2}\right\rceil, J = ⌈ log 2 n + θ log 2 κ L ⌉ , so that
K ∗ − θ ≤ n − θ e − κ L ≤ r ˉ n , log K ∗ n ≤ κ θ L + log 2. K_*^{-\theta}\le n^{-\theta}e^{-\kappa\sqrt{L}}\le\bar r_n,\qquad\log\frac{K_*}{n}\le\frac{\kappa}{\theta}\sqrt{L}+\log2. K ∗ − θ ≤ n − θ e − κ L ≤ r ˉ n , log n K ∗ ≤ θ κ L + log 2. We call level j j j high if K j > n / L 2 K_j>n/L^2 K j > n / L 2 low otherwise. The levels 0 and 1 are low. On a high level, 2 l j = K ∗ / K j < K ∗ L 2 / n 2^{l_j}=K_*/K_j<K_*L^2/n 2 l j = K ∗ / K j < K ∗ L 2 / n
R j ≤ R ˉ = A log 2 ( K ∗ L 2 / n ) + ν + 3 ≤ C L , R_j\le\bar R=A\log_2(K_*L^2/n)+\nu+3\le C\sqrt{L}, R j ≤ R ˉ = A log 2 ( K ∗ L 2 / n ) + ν + 3 ≤ C L , and there are at most log 2 ( K ∗ L 2 / n ) ≤ C L \log_2(K_*L^2/n)\le C\sqrt{L} log 2 ( K ∗ L 2 / n ) ≤ C L c L = log ( C ν R ˉ ) c_L=\log(C_\nu\bar R) c L = log ( C ν R ˉ ) 0 < c L ≤ 1 2 log L + C 0<c_L\le\frac{1}{2}\log L+C 0 < c L ≤ 2 1 log L + C
B = Δ L − 2 κ L , x j = log K j n + c L , X ˉ = log K ∗ n + c L ≤ C L . B=\Delta L-2\kappa\sqrt{L},\qquad x_j=\log\frac{K_j}{n}+c_L,\qquad\bar X=\log\frac{K_*}{n}+c_L\le C\sqrt{L}. B = Δ L − 2 κ L , x j = log n K j + c L , X ˉ = log n K ∗ + c L ≤ C L . We define the variance rule by
M j = ⌊ B x j ⌋ − ν − 2 if x j > 0 , M j = ∞ if x j ≤ 0. M_j=\left\lfloor\frac{B}{x_j}\right\rfloor-\nu-2\quad\text{if }x_j>0,\qquad M_j=\infty\quad\text{if }x_j\le0. M j = ⌊ x j B ⌋ − ν − 2 if x j > 0 , M j = ∞ if x j ≤ 0. If x j > 0 x_j>0 x j > 0 K j > n / ( C ν R ˉ ) ≥ n / L 2 K_j>n/(C_\nu\bar R)\ge n/L^2 K j > n / ( C ν R ˉ ) ≥ n / L 2 j j j 0 < x j ≤ X ˉ 0<x_j\le\bar X 0 < x j ≤ X ˉ B ≥ Δ L / 2 B\ge\Delta L/2 B ≥ Δ L /2 B / X ˉ ≥ ν + 3 B/\bar X\ge\nu+3 B / X ˉ ≥ ν + 3 M j ≥ 1 M_j\ge1 M j ≥ 1
Bias. By (49) and K ∗ − θ ≤ r ˉ n K_*^{-\theta}\le\bar r_n K ∗ − θ ≤ r ˉ n f j ( M j ) f_j(M_j) f j ( M j ) M j < ∞ M_j<\infty M j < ∞ x j > 0 x_j>0 x j > 0 M j + 1 ≤ ⌈ A l j ⌉ ≤ R ˉ M_j+1\le\lceil Al_j\rceil\le\bar R M j + 1 ≤ ⌈ A l j ⌉ ≤ R ˉ ( M j + 1 ) ν ≤ C L ν / 2 (M_j+1)^\nu\le CL^{\nu/2} ( M j + 1 ) ν ≤ C L ν /2 ρ M j ≤ e τ ( ν + 3 ) e − τ B / x j \rho^{M_j}\le e^{\tau(\nu+3)}e^{-\tau B/x_j} ρ M j ≤ e τ ( ν + 3 ) e − τ B / x j K j − θ = n − θ e θ c L e − θ x j ≤ C n − θ L θ / 2 e − θ x j K_j^{-\theta}=n^{-\theta}e^{\theta c_L}e^{-\theta x_j}\le Cn^{-\theta}L^{\theta/2}e^{-\theta x_j} K j − θ = n − θ e θ c L e − θ x j ≤ C n − θ L θ /2 e − θ x j e θ c L = ( C ν R ˉ ) θ ≤ C L θ / 2 e^{\theta c_L}=(C_\nu\bar R)^\theta\le CL^{\theta/2} e θ c L = ( C ν R ˉ ) θ ≤ C L θ /2
f j ( M j ) ≤ C n − θ L θ / 2 + ν / 2 exp ( − θ x j − τ B x j ) . f_j(M_j)\le Cn^{-\theta}L^{\theta/2+\nu/2}\exp\left(-\theta x_j-\frac{\tau B}{x_j}\right). f j ( M j ) ≤ C n − θ L θ /2 + ν /2 exp ( − θ x j − x j τ B ) . The points x j x_j x j ( 0 , X ˉ ] (0,\bar X] ( 0 , X ˉ ] log 2 \log2 log 2 Lemma 4.6 and X ˉ ≤ C L \bar X\le C\sqrt{L} X ˉ ≤ C L C n − θ L θ / 2 + ν / 2 + 1 / 4 e − 2 θ τ B Cn^{-\theta}L^{\theta/2+\nu/2+1/4}e^{-2\sqrt{\theta\tau B}} C n − θ L θ /2 + ν /2 + 1/4 e − 2 θ τ B
κ L − 2 θ τ B = 2 θ τ ( Δ L − B ) Δ L + B ≤ 4 κ θ τ Δ = 8 θ τ , \kappa\sqrt{L}-2\sqrt{\theta\tau B}
=\frac{2\sqrt{\theta\tau}(\Delta L-B)}{\sqrt{\Delta L}+\sqrt{B}}
\le\frac{4\kappa\sqrt{\theta\tau}}{\sqrt{\Delta}}=8\theta\tau, κ L − 2 θ τ B = Δ L + B 2 θ τ ( Δ L − B ) ≤ Δ 4 κ θ τ = 8 θ τ , so that the bias is at most C r ˉ n C\bar r_n C r ˉ n
Variance. Since h j ≤ 1 h_j\le1 h j ≤ 1 (45) implies that Var T ^ j , m j ≤ C ν R j n − 1 max { 1 , y j } R j − 1 \operatorname{Var}\widehat T_{j,m_j}\le C_\nu R_jn^{-1}\max\{1,y_j\}^{R_j-1} Var T j , m j ≤ C ν R j n − 1 max { 1 , y j } R j − 1 R j ≤ C L R_j\le CL R j ≤ C L K j ≤ n / L 2 K_j\le n/L^2 K j ≤ n / L 2 y j ≤ C L − 1 ≤ 1 y_j\le CL^{-1}\le1 y j ≤ C L − 1 ≤ 1 C L / n CL/n C L / n J + 1 ≤ C L J+1\le CL J + 1 ≤ C L C L 3 / 2 n − 1 / 2 CL^{3/2}n^{-1/2} C L 3/2 n − 1/2 R j ≤ R ˉ R_j\le\bar R R j ≤ R ˉ y j ≤ C ν R ˉ K j / n = e x j y_j\le C_\nu\bar R K_j/n=e^{x_j} y j ≤ C ν R ˉ K j / n = e x j x j ≤ 0 x_j\le0 x j ≤ 0 y j ≤ 1 y_j\le1 y j ≤ 1 x j > 0 x_j>0 x j > 0 R j ≤ M j + ν + 2 ≤ B / x j R_j\le M_j+\nu+2\le B/x_j R j ≤ M j + ν + 2 ≤ B / x j y j R j − 1 ≤ e R j x j ≤ e B y_j^{R_j-1}\le e^{R_jx_j}\le e^B y j R j − 1 ≤ e R j x j ≤ e B e B ≥ 1 e^B \ge1 e B ≥ 1 C ν R ˉ n − 1 e B = C ν R ˉ n − 2 θ e − 2 κ L C_\nu\bar{R}n^{-1}e^B = C_\nu\bar{R}n^{-2\theta}e^{-2\kappa\sqrt{L}} C ν R ˉ n − 1 e B = C ν R ˉ n − 2 θ e − 2 κ L C L C\sqrt{L} C L C L 3 / 4 n − θ e − κ L CL^{3/4}n^{-\theta}e^{-\kappa\sqrt{L}} C L 3/4 n − θ e − κ L ν ≥ 2 \nu\ge2 ν ≥ 2 C r ˉ n C\bar{r}_n C r ˉ n C L 3 / 2 n − 1 / 2 CL^{3/2}n^{-1/2} C L 3/2 n − 1/2 L 3 / 2 e − Δ L / 2 + κ L L^{3/2}e^{-\Delta L/2+\kappa\sqrt{L}} L 3/2 e − Δ L /2 + κ L
With λ \lambda λ Theorem 2.3 (a), and (47) follows because L θ / 2 + ν / 2 + 1 / 4 e − κ L L^{\theta/2+\nu/2+1/4}e^{-\kappa\sqrt{L}} L θ /2 + ν /2 + 1/4 e − κ L Theorem 2.3 (a). The estimator is an explicit function of the covariates X i X_i X i D D D U U U V V V W W W
It remains to prove the uniformity assertions. For compact G \mathcal{G} G g − g_- g − d g d_g d g g + g_+ g + ρ \rho ρ ( 0 , 1 ) (0,1) ( 0 , 1 ) Lemmas 4.2 and 4.3 can therefore be chosen uniformly, which gives common C 0 C_0 C 0 C ν C_\nu C ν A A A A inf G τ ≥ θ log 2 + 1 A\inf_{\mathcal{G}}\tau\ge\theta\log2+1 A inf G τ ≥ θ log 2 + 1 c K c_K c K Δ > 0 \Delta>0 Δ > 0 τ \tau τ κ \kappa κ G \mathcal{G} G R ˉ ≤ C L \bar{R}\le C\sqrt{L} R ˉ ≤ C L X ˉ ≤ C L \bar{X}\le C\sqrt{L} X ˉ ≤ C L n 1 n_1 n 1 C C C M W n − 1 / 2 M_Wn^{-1/2} M W n − 1/2 X X X ( X , Z ) (X,Z) ( X , Z )
Lower bounds
We prove Theorem 2.3 (b) in this section and the next two. The root-n n n § 6.1 ). Below the threshold, we use two fuzzy hypotheses [66 Section 2.7.4] : the prior mixtures are close in total variation, while their target means differ by more than the prior fluctuations. In a finite-response model, we bound the Hellinger distance between Poissonized mixtures (Lemma 5.2 ) and reduce the lower bound to coefficient and separation estimates (Proposition 5.3 ). Section 6 constructs priors giving c a n c a_n c a n c a n log n c a_n \log n c a n log n § 8.1 embeds them in the generic model; § 8.2 establishes localization and completes the proof.
For the construction below the threshold, we follow the strategy of the lower bound of Aronow and Lopatto [3 Section 1.2] for estimating a constant conditional variance under rough design. There, the covariate density is random under both priors, approximate moment matching makes the local laws of the data under the two priors close simultaneously for every number k ≤ M k \le M k ≤ M M ≍ log n M \asymp\sqrt{\log n} M ≍ log n e − C ( log n ) / M e^{-C(\log n)/M} e − C ( l o g n ) / M e − C log n e^{-C\sqrt{\log n}} e − C l o g n M M M M ≍ log n M \asymp\sqrt{\log n} M ≍ log n [3 ] , the hiding and the separation take a new form. Our priors share the law of a random phase Θ \Theta Θ M M M M M M κ \kappa κ log n \log n log n
Divergences and a two-point bound
For probability laws P , Q \mathbb{P},\mathbb{Q} P , Q
TV ( P , Q ) = sup E ∣ P ( E ) − Q ( E ) ∣ , d H 2 ( P , Q ) = ∫ ( d P − d Q ) 2 . \operatorname{TV}(\mathbb{P},\mathbb{Q})=\sup_{E}\lvert\mathbb{P}(E)-\mathbb{Q}(E)\rvert,\qquad d_{\mathrm{H}}^{2}(\mathbb{P},\mathbb{Q})=\int\left(\sqrt{\mathrm{d}\mathbb{P}}-\sqrt{\mathrm{d}\mathbb{Q}}\right)^{2}. TV ( P , Q ) = E sup ∣ P ( E ) − Q ( E )∣ , d H 2 ( P , Q ) = ∫ ( d P − d Q ) 2 . Any test between P \mathbb{P} P Q \mathbb{Q} Q 1 − TV ( P , Q ) 1-\operatorname{TV}(\mathbb{P},\mathbb{Q}) 1 − TV ( P , Q ) TV ≤ d H \operatorname{TV}\leq d_{\mathrm{H}} TV ≤ d H 1 − d H 2 / 2 1-d_{\mathrm{H}}^{2}/2 1 − d H 2 /2 [66 Section 2.4 and Lemma 2.3] . Since ∏ i ( 1 − x i ) ≥ 1 − ∑ i x i \prod_i(1-x_i)\geq1-\sum_i x_i ∏ i ( 1 − x i ) ≥ 1 − ∑ i x i x i ∈ [ 0 , 1 ] x_i\in[0,1] x i ∈ [ 0 , 1 ] [66 Sections 2.3 and 2.4.2] along a one-parameter path.
Lemma 5.1 (A one-dimensional testing bound). Suppose a class P \mathcal{P} P P t P_t P t t t t J J J
d P t d P ∗ = f 0 + t f 1 , f 0 + t f 1 ≥ c f > 0 , ∣ f 1 ∣ ≤ C f P ∗ -almost surely , \frac{\mathrm{d}P_t}{\mathrm{d}P_*}=f_0+tf_1,\qquad f_0+tf_1\geq c_f>0,\qquad\lvert f_1\rvert\leq C_f\quad P_*\text{-almost surely}, d P ∗ d P t = f 0 + t f 1 , f 0 + t f 1 ≥ c f > 0 , ∣ f 1 ∣ ≤ C f P ∗ -almost surely , where P ∗ P_* P ∗ c f , C f c_f,C_f c f , C f t ∈ J t\in J t ∈ J t ↦ T ( P t ) t\mapsto T(P_t) t ↦ T ( P t ) t 0 t_0 t 0 r n ( T , P ) ≥ c n − 1 / 2 r_n(T,\mathcal{P})\geq cn^{-1/2} r n ( T , P ) ≥ c n − 1/2 n n n c > 0 c>0 c > 0 n n n
Proof. Let f t = f 0 + t f 1 f_t=f_0+tf_1 f t = f 0 + t f 1
( f t − f s ) 2 = ( t − s ) 2 f 1 2 ( f t + f s ) 2 \left(\sqrt{f_t}-\sqrt{f_s}\right)^2=\frac{(t-s)^2f_1^2}{\left(\sqrt{f_t}+\sqrt{f_s}\right)^2} ( f t − f s ) 2 = ( f t + f s ) 2 ( t − s ) 2 f 1 2 and f t , f s ≥ c f f_t,f_s\geq c_f f t , f s ≥ c f d H 2 ( P t , P s ) ≤ C f 2 ( t − s ) 2 / ( 4 c f ) d_{\mathrm{H}}^2(P_t,P_s)\leq C_f^2(t-s)^2/(4c_f) d H 2 ( P t , P s ) ≤ C f 2 ( t − s ) 2 / ( 4 c f ) t ± = t 0 ± a / n t_{\pm}=t_0\pm a/\sqrt{n} t ± = t 0 ± a / n a > 0 a>0 a > 0 C f a / c f ≤ 1 / 2 C_fa/\sqrt{c_f}\leq1/2 C f a / c f ≤ 1/2 n n n J J J d H 2 d_{\mathrm{H}}^2 d H 2 d H 2 ( P t + ⊗ n , P t − ⊗ n ) ≤ C f 2 a 2 / c f ≤ 1 / 4 d_{\mathrm{H}}^2(P_{t_+}^{\otimes n},P_{t_-}^{\otimes n})\leq C_f^2a^2/c_f\leq1/4 d H 2 ( P t + ⊗ n , P t − ⊗ n ) ≤ C f 2 a 2 / c f ≤ 1/4 1 / 2 1/2 1/2 t 0 t_0 t 0 D n = ∣ T ( P t + ) − T ( P t − ) ∣ D_n=\lvert T(P_{t_+})-T(P_{t_-})\rvert D n = ∣ T ( P t + ) − T ( P t − )∣ c ′ n − 1 / 2 c'n^{-1/2} c ′ n − 1/2 c ′ > 0 c'>0 c ′ > 0 n n n 1 / 2 1/2 1/2 1 / 4 1/4 1/4 D n / 2 D_n/2 D n /2 D n / 4 D_n/4 D n /4 □ \square □
Phase priors and the Poisson comparison
Throughout the lower-bound constructions, we fix d ≥ 1 d\geq1 d ≥ 1 α , β > 0 \alpha,\beta>0 α , β > 0 0 < r − < 1 < r + < ∞ 0<r_-<1<r_+<\infty 0 < r − < 1 < r + < ∞
a ∗ = α / d , b ∗ = β / d , θ = a ∗ + b ∗ < 1 2 , a_*=\alpha/d,\qquad b_*=\beta/d,\qquad\theta=a_*+b_*<\frac{1}{2}, a ∗ = α / d , b ∗ = β / d , θ = a ∗ + b ∗ < 2 1 , ρ = r + − r − r + + r − , τ = − log ρ , δ n = ( log n ) − 2 . \rho=\frac{\sqrt{r_+}-\sqrt{r_-}}{\sqrt{r_+}+\sqrt{r_-}},\qquad\tau=-\log\rho,\qquad\delta_n=(\log n)^{-2}. ρ = r + + r − r + − r − , τ = − log ρ , δ n = ( log n ) − 2 . Here r − , r + r_-,r_+ r − , r + ρ , τ \rho,\tau ρ , τ (10) for the interval [ r − , r + ] [r_-,r_+] [ r − , r + ] r − + δ n ≤ p ≤ r + − δ n r_-+\delta_n\le p\le r_+-\delta_n r − + δ n ≤ p ≤ r + − δ n O ( δ n ) O(\delta_n) O ( δ n ) ρ M \rho^M ρ M 1 + o ( 1 ) 1+o(1) 1 + o ( 1 ) M = O ( log n ) M=O(\sqrt{\log n}) M = O ( log n ) E P [ D ∣ X ] \mathbb{E}_{P}[D\mid X] E P [ D ∣ X ] § 8.2 ). Let E \mathcal{E} E z ∈ E z\in\mathcal{E} z ∈ E
Q z ( u , v ) = q z 0 + u q z u + v q z v , q z 0 > 0 , ∑ z q z 0 = 1 , ∑ z q z u = ∑ z q z v = 0. (50) Q_z(u,v)=q_{z0}+u q_{zu}+v q_{zv},\qquad q_{z0}>0,\qquad\sum_z q_{z0}=1,\qquad\sum_z q_{zu}=\sum_z q_{zv}=0. \tag*{(50)} Q z ( u , v ) = q z 0 + u q z u + v q z v , q z 0 > 0 , z ∑ q z 0 = 1 , z ∑ q z u = z ∑ q z v = 0. (50) Here u = u ( x ) u=u(x) u = u ( x ) v = v ( x ) v=v(x) v = v ( x ) s u ( z ) = q z u / q z 0 s_u(z)=q_{zu}/q_{z0} s u ( z ) = q z u / q z 0 s v ( z ) = q z v / q z 0 s_v(z)=q_{zv}/q_{z0} s v ( z ) = q z v / q z 0 u u u v v v F F F C 4 C^4 C 4 ( 0 , 0 ) (0,0) ( 0 , 0 ) F u v ( 0 , 0 ) ≠ 0 F_{uv}(0,0)\ne0 F uv ( 0 , 0 ) = 0 T ( P ) = ∫ p F ( u , v ) T(P)=\int pF(u,v) T ( P ) = ∫ pF ( u , v ) a n = a n ( θ , τ ) a_n=a_n(\theta,\tau) a n = a n ( θ , τ ) (12) , with these θ \theta θ τ \tau τ
We next describe the priors. We fix a cube Q Q Q [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d v 0 v_0 v 0 B B B B B B d d d B / 2 B/2 B /2 ℓ ∈ { 1 , 2 } \ell\in\{1,2\} ℓ ∈ { 1 , 2 } U = [ 0 , 1 ] d \mathcal{U}=[0,1]^d U = [ 0 , 1 ] d U × { 1 , 2 } \mathcal{U}\times\{1,2\} U × { 1 , 2 } M ≥ 2 M\ge2 M ≥ 2 C 0 > 0 C_0>0 C 0 > 0 A u , A v > 0 A_u,A_v>0 A u , A v > 0 + + + − - − ( p , u , v ) (p,u,v) ( p , u , v ) [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d M M M
(A1) The parameters of distinct pairs are independent.
(A2) The function p p p [ r − , r + ] [r_-,r_+] [ r − , r + ] p 0 p_0 p 0 Q Q Q
(A3) We have u = v = 0 u=v=0 u = v = 0 Q Q Q ∣ u ∣ ≤ C 0 A u |u|\le C_0A_u ∣ u ∣ ≤ C 0 A u ∣ v ∣ ≤ C 0 A v |v|\le C_0A_v ∣ v ∣ ≤ C 0 A v
(A4) The parameters of a pair are ( Θ , H , σ u , σ v ) (\Theta,H,\sigma_u,\sigma_v) ( Θ , H , σ u , σ v ) ( Θ , H ) (\Theta,H) ( Θ , H ) Θ \Theta Θ [ 0 , 2 π ) [0,2\pi) [ 0 , 2 π ) H H H σ u , σ v ∈ { − 1 , 1 } \sigma_u,\sigma_v\in\{-1,1\} σ u , σ v ∈ { − 1 , 1 } p c , p 1 , p 2 , h u , h v p_c,p_1,p_2,h_u,h_v p c , p 1 , p 2 , h u , h v H H H Θ \Theta Θ
p = p c + p 1 cos Θ + p 2 sin Θ , p u = A u σ u h u , p v = A v σ v h v . p=p_c+p_1\cos\Theta+p_2\sin\Theta,\qquad pu=A_u\sigma_u h_u,\qquad pv=A_v\sigma_v h_v. p = p c + p 1 cos Θ + p 2 sin Θ , p u = A u σ u h u , p v = A v σ v h v . (A5) Conditional on ( Θ , H ) (\Theta,H) ( Θ , H )
P ± { σ u = s , σ v = t ∣ Θ , H } = 1 4 { 1 ± s t cos ( M Θ ) } ( s , t ∈ { − 1 , 1 } ) . P_{\pm}\{\sigma_u=s,\sigma_v=t\mid\Theta,H\}=\frac{1}{4}\{1\pm st\cos(M\Theta)\}\qquad(s,t\in\{-1,1\}). P ± { σ u = s , σ v = t ∣ Θ , H } = 4 1 { 1 ± s t cos ( M Θ )} ( s , t ∈ { − 1 , 1 }) . We write E ± E_{\pm} E ± ± \pm ± E E E ( Θ , H ) (\Theta,H) ( Θ , H ) O = ( X , Z ) O=(X,Z) O = ( X , Z ) X X X p p p P ( Z = z ∣ X = x ) = Q z ( u ( x ) , v ( x ) ) \operatorname{P}(Z = z \mid X = x) = Q_z(u(x), v(x)) P ( Z = z ∣ X = x ) = Q z ( u ( x ) , v ( x )) [49 Theorem 2.1 and Sections 3–4] , Robins et al. used priors with independent parameters on disjoint cells whose masses do not depend on the parameters, with a finite response space, to prove lower bounds for the missing-data mean and the expected conditional covariance. There the perturbations are bumps with independent random signs, and in the missing-data case the covariate density changes with the parameters without changing the mass of any cell.
By (A4), the unsigned products p u pu p u p v pv p v Θ \Theta Θ k u , k v ≥ 0 k_u,k_v \ge0 k u , k v ≥ 0 f f f ( Θ , H ) (\Theta,H) ( Θ , H )
( E + − E − ) [ σ u k u σ v k v f ( Θ , H ) ] = 21 { k u , k v o d d } E [ cos ( M Θ ) f ( Θ , H ) ] . (51) (\mathbb{E}_+ - \mathbb{E}_-)\left[\sigma_u^{k_u}\sigma_v^{k_v}f(\Theta,H)\right] = 2\mathbf{1}_{\{k_u,k_v\ \mathrm{odd}\}}\mathbb{E}\left[\cos(M\Theta)f(\Theta,H)\right]. \tag*{(51)} ( E + − E − ) [ σ u k u σ v k v f ( Θ , H ) ] = 2 1 { k u , k v odd } E [ cos ( M Θ ) f ( Θ , H ) ] . (51) In particular, each prior is invariant under ( σ u , σ v ) ↦ ( − σ u , − σ v ) (\sigma_u,\sigma_v) \mapsto(-\sigma_u,-\sigma_v) ( σ u , σ v ) ↦ ( − σ u , − σ v ) ( p , u , v ) (p,u,v) ( p , u , v ) ( p , − u , − v ) (p,-u,-v) ( p , − u , − v ) − - − + + + σ v ↦ − σ v \sigma_v \mapsto-\sigma_v σ v ↦ − σ v ( p , u , v ) (p,u,v) ( p , u , v ) ( p , u , − v ) (p,u,-v) ( p , u , − v )
As in [3 , 23 , 73 ] , we Poissonize the sample: we compare the priors through the marked Poisson process on [ 0 , 1 ] d × E [0,1]^d \times\mathcal{E} [ 0 , 1 ] d × E d x dx d x 2 n p ( x ) Q z ( u ( x ) , v ( x ) ) 2np(x)Q_z(u(x),v(x)) 2 n p ( x ) Q z ( u ( x ) , v ( x )) ∫ p = 1 \int p = 1 ∫ p = 1 p p p p ˉ = { 1 − p 0 ( 1 − v 0 ) } / v 0 \bar p = \{1-p_0(1-v_0)\}/v_0 p ˉ = { 1 − p 0 ( 1 − v 0 )} / v 0 Λ = 2 n v 0 / B \Lambda= 2nv_0/B Λ = 2 n v 0 / B μ ( d x , z ) = Λ p ˉ q z 0 d x \mu(dx,z) = \Lambda\bar p q_{z0}\,dx μ ( d x , z ) = Λ p ˉ q z 0 d x Y = ( U × { 1 , 2 } ) × E \mathcal{Y} = (\mathcal{U} \times\{1,2\}) \times\mathcal{E} Y = ( U × { 1 , 2 }) × E Λ p ( x ) Q z ( u ( x ) , v ( x ) ) \Lambda p(x)Q_z(u(x),v(x)) Λ p ( x ) Q z ( u ( x ) , v ( x )) μ \mu μ 1 + φ 1+\varphi 1 + φ
φ ( x , z ) = e 0 ( x ) + e u ( x ) s u ( z ) + e v ( x ) s v ( z ) , e 0 = p p ˉ − 1 , e u = p u p ˉ , e v = p v p ˉ . (52) \varphi(x,z) = e_0(x) + e_u(x)s_u(z) + e_v(x)s_v(z), \qquad e_0 = \frac{p}{\bar p} - 1,\quad e_u = \frac{pu}{\bar p},\quad e_v = \frac{pv}{\bar p}. \tag*{(52)} φ ( x , z ) = e 0 ( x ) + e u ( x ) s u ( z ) + e v ( x ) s v ( z ) , e 0 = p ˉ p − 1 , e u = p ˉ p u , e v = p ˉ p v . (52) Since ∑ z q z u = ∑ z q z v = 0 \sum_z q_{zu} = \sum_z q_{zv} = 0 ∑ z q z u = ∑ z q z v = 0 2 p ˉ 2\bar p 2 p ˉ ∫ φ d μ = 0 \int\varphi\,d\mu= 0 ∫ φ d μ = 0
The next lemma is in the spirit of the affinity bound of Robins et al. [49 Theorem 2.1] for two mixtures of product laws under a product prior over the cells of a partition. We use the Poisson process in place of the sample, and we expand in the factors that the points carry.
Lemma 5.2 (Comparison of the Poisson mixtures). Let two priors be admissible with frequency M M M B ≥ n B \ge n B ≥ n A u , A v ≤ 1 A_u,A_v \le1 A u , A v ≤ 1 sup x , z ( ∣ u ( x ) q z u ∣ + ∣ v ( x ) q z v ∣ ) / q z 0 ≤ 1 / 2 \sup_{x,z}(|u(x)q_{zu}| + |v(x)q_{zv}|)/q_{z0} \le1/2 sup x , z ( ∣ u ( x ) q z u ∣ + ∣ v ( x ) q z v ∣ ) / q z 0 ≤ 1/2 P B , + P o i , P B , − P o i \mathbb{P}^{\mathrm{Poi}}_{B,+},\mathbb{P}^{\mathrm{Poi}}_{B,-} P B , + Poi , P B , − Poi ± \pm ± Q Q Q j ≥ 0 j \ge0 j ≥ 0 ( U × { 1 , 2 } ) j + 2 (\mathcal{U} \times\{1,2\})^{j+2} ( U × { 1 , 2 } ) j + 2
Γ j ( x , y , ζ 1 , … , ζ j ) = ( E + − E − ) [ ( p u ) ( x ) ( p v ) ( y ) ∏ i = 1 j { p ( ζ i ) − p ˉ } ] , \Gamma_j(x,y,\zeta_1,\ldots,\zeta_j) = (\mathbb{E}_+ - \mathbb{E}_-)\left[(pu)(x)(pv)(y)\prod_{i=1}^{j}\{p(\zeta_i)-\bar p\}\right], Γ j ( x , y , ζ 1 , … , ζ j ) = ( E + − E − ) [ ( p u ) ( x ) ( p v ) ( y ) i = 1 ∏ j { p ( ζ i ) − p ˉ } ] , and let Γ ˉ j 2 \bar{\Gamma}_j^2 Γ ˉ j 2 ∥ Γ j ∥ 2 2 = ∥ Γ j ∥ L 2 ( ( U × { 1 , 2 } ) j + 2 ) 2 \|\Gamma_j\|_2^2=\|\Gamma_j\|_{L^2((U\times\{1,2\})^{j+2})}^2 ∥ Γ j ∥ 2 2 = ∥ Γ j ∥ L 2 (( U × { 1 , 2 } ) j + 2 ) 2 Γ j = 0 \Gamma_j=0 Γ j = 0 j < M j<M j < M
d H 2 ( P B , + P o i , P B , − P o i ) ≤ C B Λ 2 ∑ j ≥ M ( C Λ ) j j ! Γ ˉ j 2 + C B Λ 4 A u 2 A v 2 ( A u 2 + A v 2 ) 2 ( C Λ ) M M ! . (53) d_{\mathrm{H}}^2\left(\mathsf{P}^{\mathrm{Poi}}_{B,+},\mathsf{P}^{\mathrm{Poi}}_{B,-}\right)\le CB\Lambda^2\sum_{j\ge M}\frac{(C\Lambda)^j}{j!}\bar{\Gamma}_j^2
+CB\Lambda^4 A_u^2A_v^2(A_u^2+A_v^2)^2\frac{(C\Lambda)^M}{M!}. \tag*{(53)} d H 2 ( P B , + Poi , P B , − Poi ) ≤ C B Λ 2 j ≥ M ∑ j ! ( C Λ ) j Γ ˉ j 2 + C B Λ 4 A u 2 A v 2 ( A u 2 + A v 2 ) 2 M ! ( C Λ ) M . (53) Here C ≥ 1 C\ge1 C ≥ 1 r ± r_{\pm} r ± v 0 v_0 v 0 C 0 C_0 C 0 q z 0 , q z u , q z v q_{z0},q_{zu},q_{zv} q z 0 , q z u , q z v n n n B B B M M M A u A_u A u A v A_v A v
Proof. By (A1), each mixture is a product of mixtures over the pairs, and the squared Hellinger distance is at most the sum of the distances for the pairs. We fix a pair. Recall that a Poisson process with finite intensity measure consists of a Poisson number of independent points [30 Propositions 3.2 and 3.5] . Conditioning on the number of points and using ∫ φ d μ = 0 \int\varphi\,\mathrm{d}\mu=0 ∫ φ d μ = 0 ∏ y ∈ η { 1 + φ ( y ) } \prod_{y\in\eta}\{1+\varphi(y)\} ∏ y ∈ η { 1 + φ ( y )} η \eta η f ± ( η ) = E ± ∏ y ∈ η { 1 + φ ( y ) } f_{\pm}(\eta)=E_{\pm}\prod_{y\in\eta}\{1+\varphi(y)\} f ± ( η ) = E ± ∏ y ∈ η { 1 + φ ( y )} p / p ˉ ≥ r − / r + p/\bar p\ge r_-/r_+ p / p ˉ ≥ r − / r + 1 + φ ≥ c φ 1+\varphi\ge c_{\varphi} 1 + φ ≥ c φ c φ = r − / ( 2 r + ) c_{\varphi}=r_-/(2r_+) c φ = r − / ( 2 r + ) f ± ≥ c φ N f_{\pm}\ge c_{\varphi}^{N} f ± ≥ c φ N N N N ( f − g ) 2 = ( f − g ) 2 / ( f + g ) 2 (\sqrt{f}-\sqrt{g})^2=(f-g)^2/(\sqrt{f}+\sqrt{g})^2 ( f − g ) 2 = ( f − g ) 2 / ( f + g ) 2
d H 2 ≤ 1 4 E R [ c φ − N ( f + − f − ) 2 ] , (54) d_{\mathrm{H}}^2\le\frac{1}{4}E_{\mathrm{R}}\left[c_{\varphi}^{-N}(f_+-f_-)^2\right], \tag*{(54)} d H 2 ≤ 4 1 E R [ c φ − N ( f + − f − ) 2 ] , (54) where E R E_{\mathrm{R}} E R
We compute the right side exactly. Let ν = Π + − Π − \nu=\Pi_+-\Pi_- ν = Π + − Π − φ ω \varphi_{\omega} φ ω ω \omega ω E R ∏ y ∈ η h ( y ) = exp { ∫ ( h − 1 ) d μ } E_{\mathrm{R}}\prod_{y\in\eta}h(y)=\exp\{\int(h-1)\,\mathrm{d}\mu\} E R ∏ y ∈ η h ( y ) = exp { ∫ ( h − 1 ) d μ } h h h 0 ≤ h ≤ 1 0\le h\le1 0 ≤ h ≤ 1 [30 Section 3.3 and Exercise 3.6] . With h = c φ − 1 ( 1 + φ ω ) ( 1 + φ ω ′ ) h=c_{\varphi}^{-1}(1+\varphi_{\omega})(1+\varphi_{\omega'}) h = c φ − 1 ( 1 + φ ω ) ( 1 + φ ω ′ ) ∫ φ ω d μ = ∫ φ ω ′ d μ = 0 \int\varphi_{\omega}\,\mathrm{d}\mu=\int\varphi_{\omega'}\,\mathrm{d}\mu=0 ∫ φ ω d μ = ∫ φ ω ′ d μ = 0
E R [ c φ − N ( f + − f − ) 2 ] = e ∣ μ ∣ ( 1 / c φ − 1 ) ∬ exp { c φ − 1 ⟨ φ ω , φ ω ′ ⟩ L 2 ( μ ) } d ν ( ω ) d ν ( ω ′ ) . E_{\mathrm{R}}\left[c_{\varphi}^{-N}(f_+-f_-)^2\right]
=e^{|\mu|(1/c_{\varphi}-1)}\iint\exp\left\{c_{\varphi}^{-1}\langle\varphi_{\omega},\varphi_{\omega'}\rangle_{L^2(\mu)}\right\}\,\mathrm{d}\nu(\omega)\,\mathrm{d}\nu(\omega'). E R [ c φ − N ( f + − f − ) 2 ] = e ∣ μ ∣ ( 1/ c φ − 1 ) ∬ exp { c φ − 1 ⟨ φ ω , φ ω ′ ⟩ L 2 ( μ ) } d ν ( ω ) d ν ( ω ′ ) . We expand the exponential and use ⟨ φ ω , φ ω ′ ⟩ L 2 ( μ ) k = ⟨ φ ω ⊗ k , φ ω ′ ⊗ k ⟩ L 2 ( μ k ) \langle\varphi_{\omega},\varphi_{\omega'}\rangle_{L^2(\mu)}^k=\langle\varphi_{\omega}^{\otimes k},\varphi_{\omega'}^{\otimes k}\rangle_{L^2(\mu^k)} ⟨ φ ω , φ ω ′ ⟩ L 2 ( μ ) k = ⟨ φ ω ⊗ k , φ ω ′ ⊗ k ⟩ L 2 ( μ k ) ν \nu ν k = 0 k=0 k = 0
E R [ c φ − N ( f + − f − ) 2 ] = e ∣ μ ∣ ( 1 / c φ − 1 ) ∑ k ≥ 1 c φ − k k ! ∥ G k ∥ L 2 ( μ k ) 2 , G k = ( E + − E − ) φ ⊗ k . (55) E_{\mathrm{R}}\left[c_{\varphi}^{-N}(f_+-f_-)^2\right]
=e^{|\mu|(1/c_{\varphi}-1)}\sum_{k\ge1}\frac{c_{\varphi}^{-k}}{k!}\|G_k\|_{L^2(\mu^k)}^2,\qquad G_k=(E_+-E_-)\varphi^{\otimes k}. \tag*{(55)} E R [ c φ − N ( f + − f − ) 2 ] = e ∣ μ ∣ ( 1/ c φ − 1 ) k ≥ 1 ∑ k ! c φ − k ∥ G k ∥ L 2 ( μ k ) 2 , G k = ( E + − E − ) φ ⊗ k . (55) The interchanges are justified because the point factors are bounded and ν \nu ν ∣ μ ∣ = 2 Λ p ˉ ≤ 4 v 0 r + |\mu|=2\Lambda\bar p\le4v_0r_+ ∣ μ ∣ = 2Λ p ˉ ≤ 4 v 0 r + B ≥ n B\ge n B ≥ n c φ ˉ − N c_{\bar\varphi}^{-N} c φ ˉ − N e ∣ μ ∣ ( 1 / c φ − 1 ) e^{|\mu|(1/c_{\varphi}-1)} e ∣ μ ∣ ( 1/ c φ − 1 ) μ / c φ \mu/c_{\varphi} μ / c φ ∏ y ∈ η { 1 + φ ω ( y ) } \prod_{y\in\eta}\{1+\varphi_{\omega}(y)\} ∏ y ∈ η { 1 + φ ω ( y )} φ ω ⊗ k \varphi_{\omega}^{\otimes k} φ ω ⊗ k [31 Theorem 1.1] .
For a labeling ω ∈ { 0 , u , v } k \omega\in\{0,u,v\}^{k} ω ∈ { 0 , u , v } k k u k_u k u k v k_v k v u u u v v v j = k − k u − k v j=k-k_u-k_v j = k − k u − k v Γ ω ( x 1 , … , x k ) = ( E + − E − ) ∏ i = 1 k e ω i ( x i ) \Gamma_{\omega}(x_1,\ldots,x_k)=(E_+-E_-)\prod_{i=1}^{k}e_{\omega_i}(x_i) Γ ω ( x 1 , … , x k ) = ( E + − E − ) ∏ i = 1 k e ω i ( x i ) G k = ∑ ω s ω Γ ω G_k=\sum_{\omega}s_{\omega}\Gamma_{\omega} G k = ∑ ω s ω Γ ω s ω ( z 1 , … , z k ) s_{\omega}(z_1,\ldots,z_k) s ω ( z 1 , … , z k ) s ω i ( z i ) s_{\omega_i}(z_i) s ω i ( z i ) i i i ω i ≠ 0 \omega_i\ne0 ω i = 0 s ˉ ≥ 1 \bar s\ge1 s ˉ ≥ 1 ∣ s u ∣ |s_u| ∣ s u ∣ ∣ s v ∣ |s_v| ∣ s v ∣ 3 k 3^k 3 k ∑ z q z q 0 = 1 \sum_z q_zq_0=1 ∑ z q z q 0 = 1
∥ G k ∥ L 2 ( μ k ) 2 ≤ ( 3 s ˉ 2 Λ p ˉ ) k ∑ ω ∥ Γ ω ∥ L 2 ( ( U × { 1 , 2 } ) k ) 2 . (56) \lVert G_k\rVert_{L^2(\mu^k)}^2\le(3\bar s^2\Lambda\bar p)^k\sum_{\omega}\lVert\Gamma_{\omega}\rVert_{L^2((U\times\{1,2\})^k)}^2. \tag*{(56)} ∥ G k ∥ L 2 ( μ k ) 2 ≤ ( 3 s ˉ 2 Λ p ˉ ) k ω ∑ ∥ Γ ω ∥ L 2 (( U × { 1 , 2 } ) k ) 2 . (56) Only few labelings contribute. By (A4), we have e u = A u σ u h u / p ˉ e_u=A_u\sigma_u h_u/\bar p e u = A u σ u h u / p ˉ e v = A v σ v h v / p ˉ e_v=A_v\sigma_v h_v/\bar p e v = A v σ v h v / p ˉ ( Θ , H ) (\Theta,H) ( Θ , H ) (51) , Γ ω = 0 \Gamma_{\omega}=0 Γ ω = 0 k u k_u k u k v k_v k v Γ ω \Gamma_{\omega} Γ ω 2 A u k u A v k v p ˉ − k 2A_u^{k_u}A_v^{k_v}\bar p^{-k} 2 A u k u A v k v p ˉ − k ( Θ , H ) (\Theta,H) ( Θ , H ) cos ( M Θ ) \cos(M\Theta) cos ( M Θ ) k u + k v k_u+k_v k u + k v h u ( x i ) h_u(x_i) h u ( x i ) h v ( x i ) h_v(x_i) h v ( x i ) j j j p ( x i ) − p ˉ p(x_i)-\bar p p ( x i ) − p ˉ H H H h u , h v h_u,h_v h u , h v Θ \Theta Θ j j j j j j Θ \Theta Θ Θ \Theta Θ H H H Γ ω = 0 \Gamma_{\omega}=0 Γ ω = 0 j < M j<M j < M k u + k v k_u+k_v k u + k v
If k u = k v = 1 k_u=k_v=1 k u = k v = 1 Γ ω \Gamma_{\omega} Γ ω p ˉ − k Γ j \bar p^{-k}\Gamma_j p ˉ − k Γ j Γ j = 0 \Gamma_j=0 Γ j = 0 j < M j<M j < M k ( k − 1 ) k(k-1) k ( k − 1 ) k ( k − 1 ) / k ! = 1 / j ! k(k-1)/k!=1/j! k ( k − 1 ) / k ! = 1/ j ! p ˉ ≥ r − \bar p\ge r_- p ˉ ≥ r − C ∑ j ≥ M ( C 2 Λ ) j + 2 ∥ Γ j ∥ 2 2 / j ! C\sum_{j\ge M}(C_2\Lambda)^{j+2}\lVert\Gamma_j\rVert_2^2/j! C ∑ j ≥ M ( C 2 Λ ) j + 2 ∥ Γ j ∥ 2 2 / j ! C 2 = 3 s ˉ 2 r + / ( c φ r − 2 ) C_2=3\bar s^2r_+/(c_{\varphi}r_-^2) C 2 = 3 s ˉ 2 r + / ( c φ r − 2 ) B / 2 B/2 B /2
For the remaining labelings, q = k u + k v ≥ 4 q=k_u+k_v\ge4 q = k u + k v ≥ 4 j ≥ M j\ge M j ≥ M ∣ e 0 ∣ ≤ r + / r − |e_0|\le r_+/r_- ∣ e 0 ∣ ≤ r + / r − ∣ e u ∣ ≤ C 0 r + A u / r − |e_u|\le C_0r_+A_u/r_- ∣ e u ∣ ≤ C 0 r + A u / r − ∣ e v ∣ ≤ C 0 r + A v / r − |e_v|\le C_0r_+A_v/r_- ∣ e v ∣ ≤ C 0 r + A v / r − ( U × { 1 , 2 } ) k (U\times\{1,2\})^k ( U × { 1 , 2 } ) k 2 k 2^k 2 k ∥ Γ ω ∥ 2 2 ≤ 4 C 3 2 A u 2 k u A v 2 k v \lVert\Gamma_{\omega}\rVert_2^2\le4C_3^2A_u^{2k_u}A_v^{2k_v} ∥ Γ ω ∥ 2 2 ≤ 4 C 3 2 A u 2 k u A v 2 k v C 3 = 2 max ( 1 , C 0 ) 2 ( r + / r − ) 2 C_3=2\max(1,C_0)^2(r_+/r_-)^2 C 3 = 2 max ( 1 , C 0 ) 2 ( r + / r − ) 2 A = A u 2 + A v 2 ≤ 2 A=A_u^2+A_v^2\le2 A = A u 2 + A v 2 ≤ 2 ( q k u ) ≤ q ( q − 1 ) ( q − 2 k u − 1 ) \binom{q}{k_u}\le q(q-1)\binom{q-2}{k_u-1} ( k u q ) ≤ q ( q − 1 ) ( k u − 1 q − 2 ) 1 ≤ k u ≤ q − 1 1\le k_u\le q-1 1 ≤ k u ≤ q − 1
∑ k u = 1 q − 1 ( q k u ) A u 2 k u A v 2 ( q − k u ) ≤ q ( q − 1 ) A u 2 A v 2 A q − 2 . \sum_{k_u=1}^{q-1}\binom{q}{k_u}A_u^{2k_u}A_v^{2(q-k_u)}
\le q(q-1)A_u^2A_v^2A^{q-2}. k u = 1 ∑ q − 1 ( k u q ) A u 2 k u A v 2 ( q − k u ) ≤ q ( q − 1 ) A u 2 A v 2 A q − 2 . There are ( j + q q ) \binom{j+q}{q} ( q j + q ) u u u v v v ( j + q q ) / ( j + q ) ! = 1 / ( j ! q ! ) \binom{j+q}{q}/(j+q)!=1/(j!\,q!) ( q j + q ) / ( j + q )! = 1/ ( j ! q !) z = C 2 C 3 r − 2 Λ z=C_2C_3r_-^2\Lambda z = C 2 C 3 r − 2 Λ
C A u 2 A v 2 ( ∑ j ≥ M z j j ! ) ( ∑ q ≥ 4 z q A q − 2 ( q − 2 ) ! ) ≤ C A u 2 A v 2 A 2 z M + 4 M ! e z ( 1 + A ) CA_u^2A_v^2\left(\sum_{j\ge M}\frac{z^j}{j!}\right)\left(\sum_{q\ge4}\frac{z^qA^{q-2}}{(q-2)!}\right)
\le CA_u^2A_v^2A^2\frac{z^{M+4}}{M!}e^{z(1+A)} C A u 2 A v 2 j ≥ M ∑ j ! z j ( q ≥ 4 ∑ ( q − 2 )! z q A q − 2 ) ≤ C A u 2 A v 2 A 2 M ! z M + 4 e z ( 1 + A ) to the distance for the pair, where we used ( M + i ) ! ≥ M ! i ! (M+i)!\ge M!\,i! ( M + i )! ≥ M ! i ! ( i + 2 ) ! ≥ i ! (i+2)!\ge i! ( i + 2 )! ≥ i ! Λ ≤ 2 γ 0 \Lambda\le2\gamma_0 Λ ≤ 2 γ 0
Reduction to two estimates
The priors of Section 6 divide each block into R ≥ 1 R \ge1 R ≥ 1 ( B R ) − a ∗ (BR)^{-a_*} ( B R ) − a ∗ ( B R ) − b ∗ (BR)^{-b_*} ( B R ) − b ∗ m ≥ 1 m \ge1 m ≥ 1 R = m = 1 R=m=1 R = m = 1 (57) , R 1 − M R^{1-M} R 1 − M R ( j − M ) M R^{(j-M)\sqrt{M}} R ( j − M ) M R M R^{\sqrt{M}} R M e C E M e^{C_E M} e C E M
Proposition 5.3 (Reduction to two estimates). Under the standing assumptions of Section 5.2, let L = log n L=\log n L = log n Δ θ = 1 − 2 θ \Delta_\theta=1-2\theta Δ θ = 1 − 2 θ M M M θ Δ θ L / τ \sqrt{\theta\Delta_\theta L/\tau} θ Δ θ L / τ ρ n \rho_n ρ n ρ \rho ρ (10) computed for [ r − + δ n , r + − δ n ] [r_-+\delta_n,r_+-\delta_n] [ r − + δ n , r + − δ n ] R = R n ≥ 1 R=R_n\ge1 R = R n ≥ 1 m = m n ≥ 1 m=m_n\ge1 m = m n ≥ 1
m ≤ c ∗ M , m ≤ min ( R a ∗ , R b ∗ ) , log R = O ( log M ) m \le c_*M,\qquad m \le\min(R^{a_*},R^{b_*}),\qquad\log R=O(\log M) m ≤ c ∗ M , m ≤ min ( R a ∗ , R b ∗ ) , log R = O ( log M ) for a constant c ∗ > 0 c_*>0 c ∗ > 0 ϵ u , ϵ v ∈ ( 0 , 1 ] \epsilon_u,\epsilon_v\in(0,1] ϵ u , ϵ v ∈ ( 0 , 1 ] C 0 , C Γ ≥ 1 C_0,C_\Gamma\ge1 C 0 , C Γ ≥ 1 C E ≥ 0 C_E\ge0 C E ≥ 0 c 2 > 0 c_2>0 c 2 > 0 n n n B ≥ n B\ge n B ≥ n d d d M M M C 0 C_0 C 0
A u = ϵ u ( B R ) − a ∗ m , A v = ϵ v ( B R ) − b ∗ m , A_u=\epsilon_u(BR)^{-a_*}m,\qquad A_v=\epsilon_v(BR)^{-b_*}m, A u = ϵ u ( B R ) − a ∗ m , A v = ϵ v ( B R ) − b ∗ m , such that, for every pair and every j ≥ 0 j\ge0 j ≥ 0
∥ Γ j ∥ 2 2 ≤ C Γ j + 1 A u 2 A v 2 e C E M R 1 − M + ( j − M ) M 1 j ≥ M , (57) \lVert\Gamma_j\rVert_2^2 \le C_\Gamma^{j+1} A_u^2 A_v^2 e^{C_E M} R^{1-M+(j-M)\sqrt{M}}\mathbf{1}_{j\ge M}, \tag*{(57)} ∥ Γ j ∥ 2 2 ≤ C Γ j + 1 A u 2 A v 2 e C E M R 1 − M + ( j − M ) M 1 j ≥ M , (57) and
∣ ( E + − E − ) ∫ p w v ∣ ≥ c 2 A u A v ρ n M . (58) \left|\left(E_+-E_-\right)\int pwv\right| \ge c_2 A_u A_v \rho_n^M. \tag*{(58)} ( E + − E − ) ∫ pw v ≥ c 2 A u A v ρ n M . (58) Then, for all large n n n B B B B ≥ n B\ge n B ≥ n A u ≤ ϵ u B − a ∗ A_u\le\epsilon_uB^{-a_*} A u ≤ ϵ u B − a ∗ A v ≤ ϵ v B − b ∗ A_v\le\epsilon_vB^{-b_*} A v ≤ ϵ v B − b ∗ P P P B B B T T T P P P T ( P ) = ∫ p F ( u , v ) T(P)=\int pF(u,v) T ( P ) = ∫ pF ( u , v ) c 1 > 0 c_1>0 c 1 > 0 n n n
inf T ^ sup P ∈ P P { ∣ T ^ − T ( P ) ∣ ≥ c 1 m 2 a n } ≥ 3 8 . (59) \inf_{\widehat T}\sup_{P\in\mathcal{P}}P\left\{|\widehat T-T(P)|\ge c_1m^2a_n\right\}\ge\frac{3}{8}. \tag*{(59)} T inf P ∈ P sup P { ∣ T − T ( P ) ∣ ≥ c 1 m 2 a n } ≥ 8 3 . (59) The infimum includes randomized estimators. Writing B n B_n B n B B B B n / { n ( log n ) q } → ∞ B_n/\{n(\log n)^q\}\to\infty B n / { n ( log n ) q } → ∞ q > 0 q>0 q > 0 c 1 c_1 c 1 ϵ u ϵ v \epsilon_u\epsilon_v ϵ u ϵ v F F F
Proof. Scales. Let C ≥ 1 C\ge1 C ≥ 1 C 0 C_0 C 0 C 1 = 2 v 0 C C Γ C_1=2v_0CC_\Gamma C 1 = 2 v 0 C C Γ c 0 > 1 + log C 1 + C E c_0>1+\log C_1+C_E c 0 > 1 + log C 1 + C E
t n = Δ θ L M − log M + c 0 , (60) t_n=\frac{\Delta_\theta L}{M}-\log M+c_0, \tag*{(60)} t n = M Δ θ L − log M + c 0 , (60) we let B B B d d d k k k k d R ≥ n e t n k^dR \ge ne^{t_n} k d R ≥ n e t n x = B R / n x=BR/n x = B R / n M M M M ∗ = θ Δ θ L / τ M_*=\sqrt{\theta\Delta_\theta L/\tau} M ∗ = θ Δ θ L / τ Δ θ L = ( τ / θ ) M ∗ 2 \Delta_\theta L=(\tau/\theta)M_*^2 Δ θ L = ( τ / θ ) M ∗ 2 Δ θ L / M = ( τ / θ ) M + O ( 1 ) \Delta_\theta L/M=(\tau/\theta)M+O(1) Δ θ L / M = ( τ / θ ) M + O ( 1 ) t n = ( τ / θ ) M + O ( log M ) t_n=(\tau/\theta)M+O(\log M) t n = ( τ / θ ) M + O ( log M ) log R = O ( log M ) \log R=O(\log M) log R = O ( log M ) k ∗ = ( n e t n / R ) 1 / d k_*=(ne^{t_n}/R)^{1/d} k ∗ = ( n e t n / R ) 1/ d k ∗ ≤ k < k ∗ + 2 k_* \le k < k_*+2 k ∗ ≤ k < k ∗ + 2 1 ≤ x e − t n ≤ ( 1 + 2 / k ∗ ) d → 1 1 \le xe^{-t_n} \le(1+2/k_*)^d \to1 1 ≤ x e − t n ≤ ( 1 + 2/ k ∗ ) d → 1 x ≥ e t n x \ge e^{t_n} x ≥ e t n
log x = t n + o ( 1 ) , log ( B / n ) = τ θ M + O ( log M ) , log Λ = − τ θ M + O ( log M ) , (61) \log x=t_n+o(1),\qquad\log(B/n)=\frac{\tau}{\theta}M+O(\log M),\qquad\log\Lambda=-\frac{\tau}{\theta}M+O(\log M), \tag*{(61)} log x = t n + o ( 1 ) , log ( B / n ) = θ τ M + O ( log M ) , log Λ = − θ τ M + O ( log M ) , (61) where we used log ( B / n ) = log x − log R \log(B/n)=\log x-\log R log ( B / n ) = log x − log R Λ = 2 v 0 n / B \Lambda=2v_0n/B Λ = 2 v 0 n / B B ≥ n B\ge n B ≥ n n n n M ≍ L M\asymp\sqrt{L} M ≍ L B / { n ( log n ) q } → ∞ B/\{n(\log n)^q\}\to\infty B / { n ( log n ) q } → ∞ q > 0 q>0 q > 0 m ≤ R a ∗ m\le R^{a_*} m ≤ R a ∗ A u ≤ ϵ u B − a ∗ ≤ n − a ∗ A_u\le\epsilon_uB^{-a_*}\le n^{-a_*} A u ≤ ϵ u B − a ∗ ≤ n − a ∗ A v ≤ ϵ v B − b ∗ ≤ n − b ∗ A_v\le\epsilon_vB^{-b_*}\le n^{-b_*} A v ≤ ϵ v B − b ∗ ≤ n − b ∗ ∣ u ∣ ≤ C 0 A u |u|\le C_0A_u ∣ u ∣ ≤ C 0 A u ∣ v ∣ ≤ C 0 A v |v|\le C_0A_v ∣ v ∣ ≤ C 0 A v n n n
Indistinguishability. Let y = C C Γ Λ = C 1 n / B y=CC_\Gamma\Lambda=C_1n/B y = C C Γ Λ = C 1 n / B log y = − ( τ / θ ) M + O ( log M ) \log y=-(\tau/\theta)M+O(\log M) log y = − ( τ / θ ) M + O ( log M ) M log R = O ( M log M ) = o ( M ) \sqrt{M}\log R=O(\sqrt{M}\log M)=o(M) M log R = O ( M log M ) = o ( M ) y R M ≤ 1 / 2 yR^{\sqrt{M}}\le1/2 y R M ≤ 1/2 n n n (57) , the first term of (53) is at most
C C Γ B A u 2 A v 2 e C E M R 1 − M ∑ j ≥ M y j R ( j − M ) M j ! ≤ 2 C C Γ I n , I n = B A u 2 A v 2 R 1 − M e C E M y M M ! . CC_\Gamma BA_u^2A_v^2e^{C_EM}R^{1-M}\sum_{j\ge M}\frac{y^{j}R^{(j-M)\sqrt{M}}}{j!}\le2CC_\Gamma\mathcal{I}_n,\qquad\mathcal{I}_n=BA_u^2A_v^2R^{1-M}e^{C_EM}\frac{y^M}{M!}. C C Γ B A u 2 A v 2 e C E M R 1 − M j ≥ M ∑ j ! y j R ( j − M ) M ≤ 2 C C Γ I n , I n = B A u 2 A v 2 R 1 − M e C E M M ! y M . where we wrote j = M + i j=M+i j = M + i M ! / ( M + i ) ! ≤ 1 M!/(M+i)!\le1 M ! / ( M + i )! ≤ 1 y R M ≤ 1 / 2 yR^{\sqrt{M}}\le1/2 y R M ≤ 1/2 B Λ 2 = 4 v 0 2 n R / x B\Lambda^2=4v_0^2nR/x B Λ 2 = 4 v 0 2 n R / x ( B R ) − 2 θ = ( n x ) − 2 θ (BR)^{-2\theta}=(nx)^{-2\theta} ( B R ) − 2 θ = ( n x ) − 2 θ R 1 − M y M = R ( C 1 / x ) M R^{1-M}y^M=R(C_1/x)^M R 1 − M y M = R ( C 1 / x ) M
I n = 4 v 0 2 ( ϵ u ϵ v ) 2 n Δ θ R 2 m 4 e C E M C 1 M M ! x M + 1 + 2 θ . \mathcal{I}_n=4v_0^2(\epsilon_u\epsilon_v)^2n^{\Delta_\theta}R^2m^4e^{C_EM}\frac{C_1^M}{M!x^{M+1+2\theta}}. I n = 4 v 0 2 ( ϵ u ϵ v ) 2 n Δ θ R 2 m 4 e C E M M ! x M + 1 + 2 θ C 1 M . Using x ≥ e t n x\ge e^{t_n} x ≥ e t n M t n = Δ θ L − M log M + c 0 M Mt_n=\Delta_\theta L-M\log M+c_0M M t n = Δ θ L − M log M + c 0 M log M ! ≥ M log M − M \log M!\ge M\log M-M log M ! ≥ M log M − M
log I n ≤ log ( 4 v 0 2 ) + 2 log R + 4 log m − ( 1 + 2 θ ) t n − ( c 0 − 1 − log C 1 − C E ) M . \log\mathcal{I}_n\le\log(4v_0^2)+2\log R+4\log m-(1+2\theta)t_n-(c_0-1-\log C_1-C_E)M. log I n ≤ log ( 4 v 0 2 ) + 2 log R + 4 log m − ( 1 + 2 θ ) t n − ( c 0 − 1 − log C 1 − C E ) M . The terms 2 log R 2\log R 2 log R 4 log m 4\log m 4 log m O ( log M ) O(\log M) O ( log M ) m ≤ c ∗ M m\le c_*M m ≤ c ∗ M t n > 0 t_n>0 t n > 0 n n n c 0 c_0 c 0 I n → 0 \mathcal{I}_n\to0 I n → 0 s ∗ = min ( a ∗ , b ∗ ) s_*=\min(a_*,b_*) s ∗ = min ( a ∗ , b ∗ ) (53) , divided by I n \mathcal{I}_n I n
C Λ 2 ( A u 2 + A v 2 ) 2 R M − 1 e − C E M C Γ − M ≤ C n − 4 s ∗ e O ( M log M ) , C\Lambda^2(A_u^2+A_v^2)^2R^{M-1}e^{-C_EM}C_\Gamma^{-M}\le Cn^{-4s_*}e^{O(M\log M)}, C Λ 2 ( A u 2 + A v 2 ) 2 R M − 1 e − C E M C Γ − M ≤ C n − 4 s ∗ e O ( M l o g M ) , since Λ ≤ 2 v 0 \Lambda\le2v_0 Λ ≤ 2 v 0 A u 2 + A v 2 ≤ 2 n − 2 s ∗ A_u^2+A_v^2\le2n^{-2s_*} A u 2 + A v 2 ≤ 2 n − 2 s ∗ log R = O ( log M ) \log R=O(\log M) log R = O ( log M ) M log M = o ( L ) M\log M=o(L) M log M = o ( L ) d H ( P B , + P o i , P B , − P o i ) → 0 d_H(\mathbb{P}^{\mathrm{Poi}}_{B,+},\mathbb{P}^{\mathrm{Poi}}_{B,-})\to0 d H ( P B , + Poi , P B , − Poi ) → 0
Separation. Let D n = ∣ E + T − E − T ∣ D_n=|E_+T-E_-T| D n = ∣ E + T − E − T ∣ F o o ( u , v ) = 1 4 { F ( u , v ) − F ( − u , v ) − F ( u , − v ) + F ( − u , − v ) } F^{\mathrm{oo}}(u,v)=\frac14\{F(u,v)-F(-u,v)-F(u,-v)+F(-u,-v)\} F oo ( u , v ) = 4 1 { F ( u , v ) − F ( − u , v ) − F ( u , − v ) + F ( − u , − v )} F F F
E + T − E − T = E + ∫ p { F ( u , v ) − F ( u , − v ) } = 2 E + ∫ p F o o ( u , v ) , E_+T-E_-T=E_+\int p\{F(u,v)-F(u,-v)\}=2E_+\int pF^{\mathrm{oo}}(u,v), E + T − E − T = E + ∫ p { F ( u , v ) − F ( u , − v )} = 2 E + ∫ p F oo ( u , v ) , and in the same way ( E + − E − ) ∫ p u v = 2 E + ∫ p u v (E_+ - E_-)\int puv = 2E_+\int puv ( E + − E − ) ∫ p uv = 2 E + ∫ p uv 4 F ∞ ( u , v ) = ∫ − u u ∫ − v v F u v ( s , t ) d t d s 4F^{\infty}(u,v) = \int_{-u}^{u}\int_{-v}^{v} F_{uv}(s,t)\,\mathrm{d}t\,\mathrm{d}s 4 F ∞ ( u , v ) = ∫ − u u ∫ − v v F uv ( s , t ) d t d s u , v u,v u , v F u v F_{uv} F uv ( 0 , 0 ) (0,0) ( 0 , 0 ) F ∈ C 4 F \in C^4 F ∈ C 4 C ( s 2 + t 2 ) C(s^2+t^2) C ( s 2 + t 2 )
F ∞ ( u , v ) = F u v ( 0 , 0 ) u v + O { ∣ u v ∣ ( u 2 + v 2 ) } . F^{\infty}(u,v) = F_{uv}(0,0)uv + O\{|uv|(u^2+v^2)\}. F ∞ ( u , v ) = F uv ( 0 , 0 ) uv + O { ∣ uv ∣ ( u 2 + v 2 )} . Since ∣ u ∣ ≤ C 0 A u |u| \le C_0A_u ∣ u ∣ ≤ C 0 A u ∣ v ∣ ≤ C 0 A v |v| \le C_0A_v ∣ v ∣ ≤ C 0 A v ∫ p = 1 \int p = 1 ∫ p = 1
D n ≥ ∣ F u v ( 0 , 0 ) ∣ c 2 A u A v ρ n M − C A u A v ( A u 2 + A v 2 ) . D_n \ge|F_{uv}(0,0)|c_2A_uA_v\rho_n^M - CA_uA_v(A_u^2+A_v^2). D n ≥ ∣ F uv ( 0 , 0 ) ∣ c 2 A u A v ρ n M − C A u A v ( A u 2 + A v 2 ) . Since A u 2 + A v 2 ≤ 2 n − 2 s ∗ A_u^2+A_v^2 \le2n^{-2s_*} A u 2 + A v 2 ≤ 2 n − 2 s ∗ ρ n M = e O ( L ) \rho_n^M=e^{O(\sqrt{L})} ρ n M = e O ( L ) n n n log ρ n − log ρ = O ( δ n ) \log\rho_n-\log\rho=O(\delta_n) log ρ n − log ρ = O ( δ n ) M δ n → 0 M\delta_n\to0 M δ n → 0 ρ n M ≥ ρ M / 2 \rho_n^M\ge\rho^{M}/2 ρ n M ≥ ρ M /2 n n n f ( z ) = θ Δ θ L / z + τ z f(z)=\theta\Delta_\theta L/z+\tau z f ( z ) = θ Δ θ L / z + τ z
f ( M ) − f ( M ∗ ) = τ ( M − M ∗ ) 2 M = O ( M − 1 ) , f ( M ∗ ) = 2 θ Δ θ τ L = κ L , f(M)-f(M_*)=\frac{\tau(M-M_*)^2}{M}=O(M^{-1}),\qquad f(M_*)=2\sqrt{\theta\Delta_\theta\tau L}=\kappa\sqrt{L}, f ( M ) − f ( M ∗ ) = M τ ( M − M ∗ ) 2 = O ( M − 1 ) , f ( M ∗ ) = 2 θ Δ θ τ L = κ L , with κ \kappa κ log ( B R ) = L + t n + o ( 1 ) \log(BR)=L+t_n+o(1) log ( B R ) = L + t n + o ( 1 ) θ t n + τ M = f ( M ) − θ log M + θ c 0 \theta t_n+\tau M=f(M)-\theta\log M+\theta c_0 θ t n + τ M = f ( M ) − θ log M + θ c 0
log ( A u A v ρ M ) = log ( ϵ u ϵ v m 2 ) − θ L − f ( M ) + θ log M − θ c 0 + o ( 1 ) = log ( m 2 a n ) + O ( 1 ) , \log(A_uA_v\rho^M)=\log(\epsilon_u\epsilon_v m^2)-\theta L-f(M)+\theta\log M-\theta c_0+o(1)=\log(m^2a_n)+O(1), log ( A u A v ρ M ) = log ( ϵ u ϵ v m 2 ) − θ L − f ( M ) + θ log M − θ c 0 + o ( 1 ) = log ( m 2 a n ) + O ( 1 ) , where we used θ log M = ( θ / 2 ) log L + O ( 1 ) \theta\log M=(\theta/2)\log L+O(1) θ log M = ( θ /2 ) log L + O ( 1 ) D n ≥ c 3 m 2 a n D_n\ge c_3m^2a_n D n ≥ c 3 m 2 a n c 3 > 0 c_3>0 c 3 > 0 n n n
Testing. The target T T T C / B C/B C / B F F F Var ± T ≤ C / B \operatorname{Var}_{\pm}T\le C/B Var ± T ≤ C / B B ≥ n B\ge n B ≥ n m ≥ 1 m\ge1 m ≥ 1 θ < 1 / 2 \theta<1/2 θ < 1/2 B D n 2 ≥ c 3 2 n a n 2 = c 3 2 n 1 − 2 θ L θ e − 2 κ L → ∞ BD_n^2\ge c_3^2na_n^2=c_3^2n^{1-2\theta}L^\theta e^{-2\kappa\sqrt{L}}\to\infty B D n 2 ≥ c 3 2 n a n 2 = c 3 2 n 1 − 2 θ L θ e − 2 κ L → ∞ Q Q Q Q Q Q N N N Poisson ( 2 n ) \operatorname{Poisson}(2n) Poisson ( 2 n ) N N N P P P n n n N ≥ n N\ge n N ≥ n ( 1 − π n ) Q + ( n ) + π n δ ∗ (1-\pi_n)Q_+^{(n)}+\pi_n\delta_* ( 1 − π n ) Q + ( n ) + π n δ ∗ Q ± ( n ) Q_{\pm}^{(n)} Q ± ( n ) n n n δ ∗ \delta_* δ ∗ π n = P { Poisson ( 2 n ) < n } ≤ e − ( 1 − log 2 ) n \pi_n=P\{\operatorname{Poisson}(2n)<n\}\le e^{-(1-\log2)n} π n = P { Poisson ( 2 n ) < n } ≤ e − ( 1 − l o g 2 ) n TV ≤ d H \operatorname{TV}\le d_H TV ≤ d H TV ( Q + ( n ) , Q − ( n ) ) ≤ d H ( P B + , B − Poi , P B − , B + Poi ) / ( 1 − π n ) = o ( 1 ) \operatorname{TV}(Q_+^{(n)},Q_-^{(n)})\le d_H(\mathbb{P}_{B_+,B_-}^{\operatorname{Poi}},\mathbb{P}_{B_-,B_+}^{\operatorname{Poi}})/(1-\pi_n)=o(1) TV ( Q + ( n ) , Q − ( n ) ) ≤ d H ( P B + , B − Poi , P B − , B + Poi ) / ( 1 − π n ) = o ( 1 )
Given an estimator T ^ \widehat{T} T E ± T E_{\pm}T E ± T T ^ \widehat{T} T ± \pm ± ∣ T − E ± T ∣ < D n / 4 |T-E_{\pm}T|<D_n/4 ∣ T − E ± T ∣ < D n /4 ∣ T ^ − T ( P ) ∣ < D n / 4 |\widehat{T}-T(P)|<D_n/4 ∣ T − T ( P ) ∣ < D n /4 16 Var ± T / D n 2 16\operatorname{Var}_{\pm}T/D_n^2 16 Var ± T / D n 2 1 − TV ( Q + ( n ) , Q − ( n ) ) 1-\operatorname{TV}(Q_+^{(n)},Q_-^{(n)}) 1 − TV ( Q + ( n ) , Q − ( n ) )
sup P ∈ P P { ∣ T ^ − T ( P ) ∣ ≥ D n / 4 } ≥ 1 2 { 1 − TV ( Q + ( n ) , Q − ( n ) ) − 16 ( Var + T + Var − T ) D n 2 } = 1 2 − o ( 1 ) . \sup_{P\in\mathcal{P}}P\{|\widehat{T}-T(P)|\ge D_n/4\}\ge\frac{1}{2}\left\{1-\operatorname{TV}(Q_+^{(n)},Q_-^{(n)})-\frac{16(\operatorname{Var}_+T+\operatorname{Var}_-T)}{D_n^2}\right\}=\frac{1}{2}-o(1). P ∈ P sup P { ∣ T − T ( P ) ∣ ≥ D n /4 } ≥ 2 1 { 1 − TV ( Q + ( n ) , Q − ( n ) ) − D n 2 16 ( Var + T + Var − T ) } = 2 1 − o ( 1 ) . Since D n / 4 ≥ c 1 m 2 a n D_n/4 \ge c_1m^2a_n D n /4 ≥ c 1 m 2 a n c 1 = c 3 / 4 c_1 = c_3/4 c 1 = c 3 /4 (59) for all large n n n [66 Theorem 2.15] , and we control the prior concentration by Chebyshev’s inequality. Since an independent random seed appended to the observations does not change the total variation distance, the same argument applies to randomized estimators.
Lattice priors
We construct priors satisfying Proposition 5.3 (Lemma 6.2 ). In each pair, the density on the bump supports is r 0 ± h n cos ( Θ + ϕ ( x ) ) r_0 \pm h_n \cos(\Theta+\phi(x)) r 0 ± h n cos ( Θ + ϕ ( x )) Θ \Theta Θ ϕ \phi ϕ S ∈ [ 0 , 1 ] S \in[0,1] S ∈ [ 0 , 1 ] ± cos ( M Θ ) \pm\cos(M\Theta) ± cos ( M Θ ) ϕ \phi ϕ ξ a , q \xi_{a,q} ξ a , q a = 1 , … , J a=1,\ldots,J a = 1 , … , J q = 1 , … , d q=1,\ldots,d q = 1 , … , d J = 0 J=0 J = 0 ϕ = 0 \phi=0 ϕ = 0 S = 1 S=1 S = 1 c a n c\,a_n c a n J ≥ 1 J\geq1 J ≥ 1 log n \log n log n
(i) Outside a set of small measure, ϕ ∈ ( 2 π / M ) Z \phi\in(2\pi/M)\mathbb{Z} ϕ ∈ ( 2 π / M ) Z M M M 1 / p 1/p 1/ p Θ \Theta Θ ρ n M \rho_n^M ρ n M
(ii) The gated characteristic functions of ξ a , q \xi_{a,q} ξ a , q ⋃ l ∈ Z [ l M − K a , l M + K a ] \bigcup_{l\in\mathbb{Z}}[lM-K_a,lM+K_a] ⋃ l ∈ Z [ l M − K a , l M + K a ] u u u v v v j j j Θ \Theta Θ j < M j<M j < M M ≤ j < M + M M\leq j<M+\sqrt{M} M ≤ j < M + M R 1 − M e C E M R^{1-M}e^{C_E M} R 1 − M e C E M R R R j j j
(iii) The gate controls weighted coefficient powers: S S S ϕ \phi ϕ O ( 1 / m ) O(1/m) O ( 1/ m ) m ≍ M m\asymp M m ≍ M 1 / m 1/m 1/ m m ( B R ) − a ∗ m(BR)^{-a_*} m ( B R ) − a ∗ m ( B R ) − b ∗ m(BR)^{-b_*} m ( B R ) − b ∗
Lattice laws with compact Fourier support
Throughout this section we fix the integer Q = ⌈ max ( α , β ) ⌉ + 2 Q=\lceil\max(\alpha,\beta)\rceil+2 Q = ⌈ max ( α , β )⌉ + 2
Π ( y ) = c Q − 1 ( sin y y ) 2 Q , c Q = ∫ R ( sin y y ) 2 Q d y . \Pi(y)=c_Q^{-1}\left(\frac{\sin y}{y}\right)^{2Q},\qquad c_Q=\int_{\mathbb{R}}\left(\frac{\sin y}{y}\right)^{2Q}\,dy. Π ( y ) = c Q − 1 ( y sin y ) 2 Q , c Q = ∫ R ( y sin y ) 2 Q d y . where sin y / y \sin y/y sin y / y y = 0 y=0 y = 0 Y ∼ Π Y\sim\Pi Y ∼ Π μ 2 = E Y 2 \mu_2=EY^2 μ 2 = E Y 2 Π ( y ) ≤ c Q − 1 min ( 1 , ∣ y ∣ − 2 Q ) \Pi(y)\leq c_Q^{-1}\min(1,|y|^{-2Q}) Π ( y ) ≤ c Q − 1 min ( 1 , ∣ y ∣ − 2 Q ) f ^ ( w ) = ∫ R f ( x ) e i w x d x \widehat{f}(w)=\int_{\mathbb{R}}f(x)e^{iwx}\,dx f ( w ) = ∫ R f ( x ) e i w x d x
Lemma 6.1 (Gated lattice laws). Let η > 0 \eta>0 η > 0 λ ≥ 1 \lambda\geq1 λ ≥ 1 M ≥ 2 M\geq2 M ≥ 2 h M = 2 π / M h_M=2\pi/M h M = 2 π / M π η ( x ) = η − 1 Π ( x / η ) \pi_\eta(x)=\eta^{-1}\Pi(x/\eta) π η ( x ) = η − 1 Π ( x / η ) K = 6 Q / η K=6Q/\eta K = 6 Q / η
G ( x ) = ( sin ( x / ( λ η ) ) x / ( λ η ) ) 2 Q ( x ∈ R ) , G ( 0 ) = 1. G(x)=\left(\frac{\sin(x/(\lambda\eta))}{x/(\lambda\eta)}\right)^{2Q}\qquad(x\in\mathbb{R}),\qquad G(0)=1. G ( x ) = ( x / ( λ η ) sin ( x / ( λ η )) ) 2 Q ( x ∈ R ) , G ( 0 ) = 1. If 4 K ≤ M 4K \le M 4 K ≤ M h M π η ( x ) h_M\pi_\eta(x) h M π η ( x ) x ∈ h M Z x \in h_M\mathbb{Z} x ∈ h M Z ξ \xi ξ P { ξ = x } = h M π η ( x ) P\{\xi= x\} = h_M\pi_\eta(x) P { ξ = x } = h M π η ( x ) h M Z h_M\mathbb{Z} h M Z
0 ≤ G ≤ 1 , 1 − E G ( ξ ) ≤ Q μ 2 3 λ 2 , G ( x ) ∣ x ∣ r ≤ ( λ η ) r ( x ∈ R , 0 ≤ r ≤ 2 Q − 2 ) . (62) 0 \le G \le1,\qquad1 - EG(\xi) \le\frac{Q\mu_2}{3\lambda^2},\qquad G(x)|x|^r \le(\lambda\eta)^r \qquad(x \in\mathbb{R},\ 0 \le r \le2Q - 2). \tag*{(62)} 0 ≤ G ≤ 1 , 1 − E G ( ξ ) ≤ 3 λ 2 Q μ 2 , G ( x ) ∣ x ∣ r ≤ ( λ η ) r ( x ∈ R , 0 ≤ r ≤ 2 Q − 2 ) . (62) Moreover, the function Ξ ^ ( w ) = E [ G ( ξ ) e i w ξ ~ ] \widehat{\Xi}(w) = E[G(\xi)e^{iw\widetilde{\xi}}] Ξ ( w ) = E [ G ( ξ ) e i w ξ ] w ∈ R w \in\mathbb{R} w ∈ R ∣ Ξ ^ ∣ ≤ 1 |\widehat{\Xi}| \le1 ∣ Ξ ∣ ≤ 1 Ξ ^ ( w ) = 0 \widehat{\Xi}(w) = 0 Ξ ( w ) = 0 ∣ w − l M ∣ ≤ K |w-lM| \le K ∣ w − l M ∣ ≤ K l ∈ Z l \in\mathbb{Z} l ∈ Z
Proof. The identity sin y / y = 1 2 ∫ − 1 1 e i ω y d ω \sin y/y = \frac{1}{2}\int_{-1}^{1}e^{i\omega y}\,d\omega sin y / y = 2 1 ∫ − 1 1 e iω y d ω ( sin y / y ) 2 Q (\sin y/y)^{2Q} ( sin y / y ) 2 Q 2 Q 2Q 2 Q 1 2 1 [ − 1 , 1 ] \frac{1}{2}\mathbf{1}_{[-1,1]} 2 1 1 [ − 1 , 1 ] [ − 2 Q , 2 Q ] [-2Q,2Q] [ − 2 Q , 2 Q ] π ^ η \widehat{\pi}_\eta π η G ^ \widehat{G} G [ − 2 Q / η , 2 Q / η ] [-2Q/\eta,2Q/\eta] [ − 2 Q / η , 2 Q / η ] [ − 2 Q / ( λ η ) , 2 Q / ( λ η ) ] [-2Q/(\lambda\eta),2Q/(\lambda\eta)] [ − 2 Q / ( λ η ) , 2 Q / ( λ η )] λ ≥ 1 \lambda\ge1 λ ≥ 1 G k π η ^ \widehat{G^k\pi_\eta} G k π η [ − K , K ] [-K,K] [ − K , K ] k = 0 , 1 , 2 k=0,1,2 k = 0 , 1 , 2
Let f = G k π η f=G^k\pi_\eta f = G k π η k ∈ { 0 , 1 , 2 } k\in\{0,1,2\} k ∈ { 0 , 1 , 2 } w ∈ R w\in\mathbb{R} w ∈ R 0 ≤ f ( x ) ≤ C η ( 1 + ∣ x ∣ ) − 2 Q 0\le f(x)\le C_\eta(1+|x|)^{-2Q} 0 ≤ f ( x ) ≤ C η ( 1 + ∣ x ∣ ) − 2 Q h M h_M h M y ↦ ∑ x ∈ h M Z f ( y + x ) e i w ( y + x ) y\mapsto\sum_{x\in h_M\mathbb{Z}}f(y+x)e^{iw(y+x)} y ↦ ∑ x ∈ h M Z f ( y + x ) e i w ( y + x ) h M − 1 f ^ ( w − l M ) h_M^{-1}\widehat{f}(w-lM) h M − 1 f ( w − l M ) l ∈ Z l\in\mathbb{Z} l ∈ Z y = 0 y=0 y = 0 [61 Chapter 5, Section 3] ,
h M ∑ x ∈ h M Z f ( x ) e i w x = ∑ l ∈ Z f ^ ( w + l M ) . (63) h_M\sum_{x\in h_M\mathbb{Z}}f(x)e^{iwx}=\sum_{l\in\mathbb{Z}}\widehat{f}(w+lM). \tag*{(63)} h M x ∈ h M Z ∑ f ( x ) e i w x = l ∈ Z ∑ f ( w + l M ) . (63) At w = 0 w=0 w = 0 l = 0 l=0 l = 0 K < M K<M K < M k = 0 k=0 k = 0 h M π η ( x ) h_M\pi_\eta(x) h M π η ( x ) π ^ η ( 0 ) = 1 \widehat{\pi}_\eta(0)=1 π η ( 0 ) = 1 k = 1 k=1 k = 1 E G ( ξ ~ ) = ∫ G π η EG(\widetilde{\xi})=\int G\pi_\eta E G ( ξ ) = ∫ G π η k = 2 k=2 k = 2 (63) gives Ξ ^ ( w ) = ∑ l f ^ ( w + l M ) \widehat{\Xi}(w)=\sum_l\widehat{f}(w+lM) Ξ ( w ) = ∑ l f ( w + l M ) f ^ \widehat{f} f [ − K , K ] [-K,K] [ − K , K ] 2 K < M 2K<M 2 K < M ∣ w + l M ∣ ≤ K |w+lM|\le K ∣ w + l M ∣ ≤ K ∣ Ξ ^ ∣ ≤ E G ( ξ ~ ) 2 ≤ 1 |\widehat{\Xi}|\le EG(\widetilde{\xi})^2\le1 ∣ Ξ ∣ ≤ E G ( ξ ) 2 ≤ 1
The bound ∣ sin z / z ∣ ≤ min ( 1 , ∣ z ∣ − 1 ) |\sin z/z|\le\min(1,|z|^{-1}) ∣ sin z / z ∣ ≤ min ( 1 , ∣ z ∣ − 1 ) 0 ≤ G ≤ 1 0\le G\le1 0 ≤ G ≤ 1 G ( x ) ∣ x ∣ r ≤ ( λ η ) r G(x)|x|^r\le(\lambda\eta)^r G ( x ) ∣ x ∣ r ≤ ( λ η ) r 0 ≤ r ≤ 2 Q 0\le r\le2Q 0 ≤ r ≤ 2 Q U , V U,V U , V [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] ( sin z / z ) 2 = E cos ( z ( U + V ) ) ≥ 1 − z 2 E ( U + V ) 2 / 2 = 1 − z 2 / 3 (\sin z/z)^2=E\cos(z(U+V))\ge1-z^2E(U+V)^2/2=1-z^2/3 ( sin z / z ) 2 = E cos ( z ( U + V )) ≥ 1 − z 2 E ( U + V ) 2 /2 = 1 − z 2 /3 1 − ( sin z / z ) 2 Q ≤ Q { 1 − ( sin z / z ) 2 } ≤ Q z 2 / 3 1-(\sin z/z)^{2Q}\le Q\{1-(\sin z/z)^2\}\le Qz^2/3 1 − ( sin z / z ) 2 Q ≤ Q { 1 − ( sin z / z ) 2 } ≤ Q z 2 /3 ∫ G π η = E ( sin ( Y / λ ) / ( Y / λ ) ) 2 Q \int G\pi_\eta=E(\sin(Y/\lambda)/(Y/\lambda))^{2Q} ∫ G π η = E ( sin ( Y / λ ) / ( Y / λ ) ) 2 Q Y ∼ Π Y\sim\Pi Y ∼ Π 1 − E G ( ξ ~ ) ≤ Q μ 2 / ( 3 λ 2 ) 1-EG(\widetilde{\xi})\le Q\mu_2/(3\lambda^2) 1 − E G ( ξ ) ≤ Q μ 2 / ( 3 λ 2 )
Lattice laws obtained in this way, by sampling a probability density whose Fourier transform has compact support, are constructed in [68 Section V, Proposition 5 and Corollary 6] . There, the sampled masses are normalized by the same formula, the characteristic function is the sum of the translated copies of the Fourier transform, and Example 1 uses powers of sin y / y \sin y/y sin y / y
Construction of the priors
We use the cube Q \mathcal{Q} Q v 0 v_0 v 0 x ∈ U = [ 0 , 1 ] d x\in\mathcal{U}=[0,1]^d x ∈ U = [ 0 , 1 ] d ι i n , ι o u t : U → [ 0 , 1 ] \iota_{\mathrm{in}},\iota_{\mathrm{out}}:\mathcal{U}\to[0,1] ι in , ι out : U → [ 0 , 1 ] U \mathcal{U} U ι i n \iota_{\mathrm{in}} ι in ι o u t = 1 \iota_{\mathrm{out}}=1 ι out = 1 ι i n \iota_{\mathrm{in}} ι in R d \mathbb{R}^{d} R d ι ˉ = ∫ U ι o u t \bar{\iota}=\int_{\mathcal{U}}\iota_{\mathrm{out}} ι ˉ = ∫ U ι out I i n = ∫ U ι i n 2 > 0 I_{\mathrm{in}}=\int_{\mathcal{U}}\iota_{\mathrm{in}}^{2}>0 I in = ∫ U ι in 2 > 0
r 0 = r − + r + 2 , h n = r + − r − 2 − δ n , p 0 = 1 − v 0 ι ˉ r 0 1 − v 0 ι ˉ . r_0=\frac{r_-+r_+}{2},\qquad h_n=\frac{r_+-r_-}{2}-\delta_n,\qquad p_0=\frac{1-v_0\bar{\iota}r_0}{1-v_0\bar{\iota}}. r 0 = 2 r − + r + , h n = 2 r + − r − − δ n , p 0 = 1 − v 0 ι ˉ 1 − v 0 ι ˉ r 0 . and we choose v 0 v_0 v 0 p 0 p_0 p 0 ( r − , r + ) (r_-,r_+) ( r − , r + ) n n n
0 < δ n ≤ 1 2 min { r 0 − r − , p 0 − r − , r + − p 0 } . 0<\delta_n\leq\frac{1}{2}\min\{r_0-r_-,p_0-r_-,r_+-p_0\}. 0 < δ n ≤ 2 1 min { r 0 − r − , p 0 − r − , r + − p 0 } . We write s 0 = min ( α , β ) s_0=\min(\alpha,\beta) s 0 = min ( α , β ) α 0 = min ( s 0 , 1 ) / 2 \alpha_0=\min(s_0,1)/2 α 0 = min ( s 0 , 1 ) /2 0 < α 0 < 1 0<\alpha_0<1 0 < α 0 < 1 α 0 < s 0 \alpha_0<s_0 α 0 < s 0 J ≥ 0 J\geq0 J ≥ 0 R = 2 J d R=2^{Jd} R = 2 J d m = 2 J s 0 m=2^{Js_0} m = 2 J s 0 R R R 2 − J 2^{-J} 2 − J
Digit selectors. For y ∈ R y\in\mathbb{R} y ∈ R a ≥ 1 a\geq1 a ≥ 1 e a ( y ) = ⌊ 2 a y ⌋ m o d 2 e_a(y)=\lfloor2^a y\rfloor\bmod2 e a ( y ) = ⌊ 2 a y ⌋ mod 2 x x x U \mathcal{U} U e a ( x q ) e_a(x_q) e a ( x q ) 1 ≤ a ≤ J 1\leq a\leq J 1 ≤ a ≤ J 1 ≤ q ≤ d 1\leq q\leq d 1 ≤ q ≤ d { 0 , 1 } \{0,1\} { 0 , 1 } d J dJ dJ x x x C ∞ C^\infty C ∞ Υ \Upsilon Υ Υ = 0 \Upsilon=0 Υ = 0 ( − ∞ , − 1 ] (-\infty,-1] ( − ∞ , − 1 ] Υ = 1 \Upsilon=1 Υ = 1 [ 1 , ∞ ) [1,\infty) [ 1 , ∞ ) Υ ( 0 ) = 1 / 2 \Upsilon(0)=1/2 Υ ( 0 ) = 1/2 0 < γ ≤ 1 / 4 0<\gamma\leq1/4 0 < γ ≤ 1/4 y ∈ R y\in\mathbb{R} y ∈ R z z z y y y
ϖ γ ( y ) = Υ ( ( − 1 ) z + 1 ( y − z ) / γ ) . \varpi_\gamma(y)=\Upsilon\left((-1)^{z+1}(y-z)/\gamma\right). ϖ γ ( y ) = Υ ( ( − 1 ) z + 1 ( y − z ) / γ ) . At distance at least γ \gamma γ ϖ γ ( y ) \varpi_\gamma(y) ϖ γ ( y ) ⌊ y ⌋ m o d 2 \lfloor y\rfloor\bmod2 ⌊ y ⌋ mod 2 z z z ϖ γ \varpi_\gamma ϖ γ C ∞ C^\infty C ∞ [ 0 , 1 ] [0,1] [ 0 , 1 ] ∣ ϖ γ ( N ) ∣ ≤ C N γ − N |\varpi_\gamma^{(N)}|\leq C_N\gamma^{-N} ∣ ϖ γ ( N ) ∣ ≤ C N γ − N N ≥ 1 N\geq1 N ≥ 1 ⌊ y ⌋ \lfloor y\rfloor ⌊ y ⌋ z = ⌊ y ⌋ z=\lfloor y\rfloor z = ⌊ y ⌋ y − z ≥ 0 y-z\geq0 y − z ≥ 0 z = ⌊ y ⌋ + 1 z=\lfloor y\rfloor+1 z = ⌊ y ⌋ + 1 z − y > 0 z-y>0 z − y > 0 ϖ γ ( y ) ≥ Υ ( 0 ) = 1 / 2 \varpi_\gamma(y)\geq\Upsilon(0)=1/2 ϖ γ ( y ) ≥ Υ ( 0 ) = 1/2 ϖ γ ( y ) ≤ 1 / 2 \varpi_\gamma(y)\leq1/2 ϖ γ ( y ) ≤ 1/2 ⌊ y ⌋ \lfloor y\rfloor ⌊ y ⌋
∣ ϖ γ ( y ) − e ∣ ≥ 1 2 if e ∈ { 0 , 1 } and e ≠ ⌊ y ⌋ m o d 2. (64) |\varpi_\gamma(y)-e|\geq\frac{1}{2}\qquad\text{if }e\in\{0,1\}\text{ and }e\ne\lfloor y\rfloor\bmod2. \tag*{(64)} ∣ ϖ γ ( y ) − e ∣ ≥ 2 1 if e ∈ { 0 , 1 } and e = ⌊ y ⌋ mod 2. (64) Scales. For a level 1 ≤ a ≤ J 1\leq a\leq J 1 ≤ a ≤ J b = J − a b=J-a b = J − a
γ a = γ ∗ ( b + 1 ) 2 , λ a = λ ∗ ( b + 1 ) , η a = γ a 2 b α 0 λ a m , K a = 6 Q η a , \gamma_a=\frac{\gamma_*}{(b+1)^2},\qquad\lambda_a=\lambda_*(b+1),\qquad\eta_a=\frac{\gamma_a2^{b\alpha_0}}{\lambda_a m},\qquad K_a=\frac{6Q}{\eta_a}, γ a = ( b + 1 ) 2 γ ∗ , λ a = λ ∗ ( b + 1 ) , η a = λ a m γ a 2 b α 0 , K a = η a 6 Q , where γ ∗ = min { 1 / 4 , I i n / ( 16 d ) } \gamma_*=\min\{1/4,I_{\mathrm{in}}/(16d)\} γ ∗ = min { 1/4 , I in / ( 16 d )} λ ∗ = max { 1 , 8 Q d μ 2 / 3 } \lambda_*=\max\{1,\sqrt{8Qd\mu_2/3}\} λ ∗ = max { 1 , 8 Q d μ 2 /3 } λ a η a = γ a 2 b α 0 / m \lambda_a\eta_a=\gamma_a2^{b\alpha_0}/m λ a η a = γ a 2 b α 0 / m K a = K ( b ) m K_a=K(b)m K a = K ( b ) m K ( b ) = 6 Q λ ∗ ( b + 1 ) 3 2 − b α 0 / γ ∗ K(b)=6Q\lambda_*(b+1)^3 2^{-b\alpha_0}/\gamma_* K ( b ) = 6 Q λ ∗ ( b + 1 ) 3 2 − b α 0 / γ ∗ K ˉ = sup b ≥ 0 K ( b ) \bar{K}=\sup_{b\geq0}K(b) K ˉ = sup b ≥ 0 K ( b ) c ∗ = 1 / ( 4 K ˉ ) c_* =1/(4\bar{K}) c ∗ = 1/ ( 4 K ˉ ) m ≤ c ∗ M m\leq c_*M m ≤ c ∗ M
4 K a ≤ M ( 1 ≤ a ≤ J ) . (65) 4K_a\leq M\qquad(1\leq a\leq J). \tag*{(65)} 4 K a ≤ M ( 1 ≤ a ≤ J ) . (65) All these constants depend only on d d d α \alpha α β \beta β ι i n \iota_{\mathrm{in}} ι in M M M
The phase and the gate. Let M ≥ 2 M \ge2 M ≥ 2 m ≤ c ∗ M m \le c_*M m ≤ c ∗ M h M = 2 π / M h_M = 2\pi/M h M = 2 π / M ξ a , q \xi_{a,q} ξ a , q 1 ≤ a ≤ J 1 \le a \le J 1 ≤ a ≤ J 1 ≤ q ≤ d 1 \le q \le d 1 ≤ q ≤ d ξ \xi ξ η = η a \eta= \eta_a η = η a λ = λ a \lambda= \lambda_a λ = λ a G a G_a G a Ξ a \Xi_a Ξ a G G G Ξ \Xi Ξ
S = ∏ a , q G a ( ξ a , q ) , s a , q ( x ) = ϖ γ a ( 2 a x q ) , ϕ ( x ) = ∑ a = 1 J ∑ q = 1 d ξ a , q s a , q ( x ) S = \prod_{a,q} G_a(\xi_{a,q}), \qquad s_{a,q}(x) = \varpi_{\gamma_a}(2^a x_q), \qquad\phi(x) = \sum_{a=1}^{J} \sum_{q=1}^{d} \xi_{a,q}s_{a,q}(x) S = a , q ∏ G a ( ξ a , q ) , s a , q ( x ) = ϖ γ a ( 2 a x q ) , ϕ ( x ) = a = 1 ∑ J q = 1 ∑ d ξ a , q s a , q ( x ) for x ∈ R d x \in\mathbb{R}^d x ∈ R d ∏ i ( 1 − t i ) ≥ 1 − ∑ i t i \prod_i(1-t_i) \ge1-\sum_i t_i ∏ i ( 1 − t i ) ≥ 1 − ∑ i t i t i ∈ [ 0 , 1 ] t_i \in[0,1] t i ∈ [ 0 , 1 ]
E S ≥ 1 − Q d μ 2 3 λ ∗ 2 ∑ b ≥ 0 ( b + 1 ) − 2 ≥ 3 4 , E S 2 ≥ ( E S ) 2 ≥ 9 16 . \mathbb{E}S \ge1-\frac{Qd\mu_2}{3\lambda_*^2}\sum_{b\ge0}(b+1)^{-2} \ge\frac{3}{4}, \qquad\mathbb{E}S^2 \ge(\mathbb{E}S)^2 \ge\frac{9}{16}. E S ≥ 1 − 3 λ ∗ 2 Q d μ 2 b ≥ 0 ∑ ( b + 1 ) − 2 ≥ 4 3 , E S 2 ≥ ( E S ) 2 ≥ 16 9 . where we used ∑ b ≥ 0 ( b + 1 ) − 2 < 2 \sum_{b\ge0}(b+1)^{-2}<2 ∑ b ≥ 0 ( b + 1 ) − 2 < 2 λ ∗ \lambda_* λ ∗ S ≤ G a ( ξ a , q ) S \le G_a(\xi_{a,q}) S ≤ G a ( ξ a , q ) ( a , q ) (a,q) ( a , q ) 0 ≤ G a ≤ 1 0 \le G_a \le1 0 ≤ G a ≤ 1 ( a i , q i ) (a_i,q_i) ( a i , q i ) r i ≥ 1 r_i \ge1 r i ≥ 1 ∑ i r i ≤ 2 Q − 2 \sum_i r_i \le2Q-2 ∑ i r i ≤ 2 Q − 2
S ∏ i ∣ ξ a i , q i ∣ r i ≤ ∏ i ( λ a i η a i ) r i . (66) S\prod_i\lvert\xi_{a_i,q_i}\rvert^{r_i} \le\prod_i(\lambda_{a_i}\eta_{a_i})^{r_i}. \tag*{(66)} S i ∏ ∣ ξ a i , q i ∣ r i ≤ i ∏ ( λ a i η a i ) r i . (66) Let U ∘ \mathcal{U}^{\circ} U ∘ x ∈ U x \in\mathcal{U} x ∈ U 2 a x q 2^a x_q 2 a x q γ a \gamma_a γ a a a a q q q U ∘ \mathcal{U}^{\circ} U ∘ s a , q ( x ) = e a ( x q ) ∈ { 0 , 1 } s_{a,q}(x)=e_a(x_q)\in\{0,1\} s a , q ( x ) = e a ( x q ) ∈ { 0 , 1 } ϕ ( x ) \phi(x) ϕ ( x ) h M Z h_M\mathbb{Z} h M Z
cos ( M ϕ ( x ) ) = 1 ( x ∈ U ∘ ) . (67) \cos(M\phi(x))=1 \qquad(x\in\mathcal{U}^{\circ}). \tag*{(67)} cos ( M ϕ ( x )) = 1 ( x ∈ U ∘ ) . (67) For each a a a q q q x ∈ U x\in\mathcal{U} x ∈ U 2 a x q 2^a x_q 2 a x q γ a \gamma_a γ a 2 γ a 2\gamma_a 2 γ a ∑ a γ a < 2 γ ∗ \sum_a\gamma_a<2\gamma_* ∑ a γ a < 2 γ ∗
∣ U ∖ U ∘ ∣ < 4 d γ ∗ ≤ I i n / 4. (68) \lvert\mathcal{U}\setminus\mathcal{U}^{\circ}\rvert<4d\gamma_*\le I_{\mathrm{in}}/4. \tag*{(68)} ∣ U ∖ U ∘ ∣ < 4 d γ ∗ ≤ I in /4. (68) The priors. In each pair, let Θ \Theta Θ [ 0 , 2 π ) [0,2\pi) [ 0 , 2 π ) ξ ~ = ( ξ a , q ) a , q \tilde{\xi}=(\xi_{a,q})_{a,q} ξ ~ = ( ξ a , q ) a , q ( Θ , ξ ~ ) (\Theta,\tilde{\xi}) ( Θ , ξ ~ ) ℓ \ell ℓ
r ( x ) = r 0 + ( − 1 ) ℓ − 1 h n cos ( Θ + ϕ ( x ) ) , p = p 0 + ι o u t ( r − p 0 ) , r(x)=r_0+(-1)^{\ell-1}h_n\cos(\Theta+\phi(x)), \qquad p=p_0+\iota_{\mathrm{out}}(r-p_0), r ( x ) = r 0 + ( − 1 ) ℓ − 1 h n cos ( Θ + ϕ ( x )) , p = p 0 + ι out ( r − p 0 ) , and p = p 0 p=p_0 p = p 0 Q Q Q ϕ \phi ϕ C ∞ C^\infty C ∞ ϕ \phi ϕ ∫ p \int p ∫ p p p p 2 ( v 0 / B ) p ˉ 2(v_0/B)\bar p 2 ( v 0 / B ) p ˉ
p ˉ = p 0 + ι ˉ ( r 0 − p 0 ) \bar p=p_0+\bar{\iota}(r_0-p_0) p ˉ = p 0 + ι ˉ ( r 0 − p 0 ) is the mean value of p p p p 0 ( 1 − v 0 ) + v 0 p ˉ = p 0 + v 0 ι ˉ ( r 0 − p 0 ) = 1 p_0(1-v_0)+v_0\bar p=p_0+v_0\bar{\iota}(r_0-p_0)=1 p 0 ( 1 − v 0 ) + v 0 p ˉ = p 0 + v 0 ι ˉ ( r 0 − p 0 ) = 1 ι o u t \iota_{\mathrm{out}} ι out p p p C ∞ C^\infty C ∞ ι o u t \iota_{\mathrm{out}} ι out p 0 p_0 p 0 r ∈ [ r − + δ n , r + − δ n ] r \in[r_-+\delta_n,r_+-\delta_n] r ∈ [ r − + δ n , r + − δ n ] r − + δ n ≤ p ≤ r + − δ n r_-+\delta_n \le p \le r_+-\delta_n r − + δ n ≤ p ≤ r + − δ n
Conditional on the density parameters, the signs σ u , σ v ∈ { − 1 , 1 } \sigma_u,\sigma_v \in\{-1,1\} σ u , σ v ∈ { − 1 , 1 }
P ± { σ u = s , σ v = t ∣ Θ , ξ } = 1 4 { 1 ± s t cos ( M Θ ) } ( s , t ∈ { − 1 , 1 } ) P_{\pm}\{\sigma_u=s,\sigma_v=t\mid\Theta,\xi\}=\frac{1}{4}\{1\pm st\cos(M\Theta)\}\qquad(s,t\in\{-1,1\}) P ± { σ u = s , σ v = t ∣ Θ , ξ } = 4 1 { 1 ± s t cos ( M Θ )} ( s , t ∈ { − 1 , 1 }) under the ± \pm ± H = ξ H=\xi H = ξ § 6.2 , E E E ϵ u , ϵ v ∈ ( 0 , 1 ] \epsilon_u,\epsilon_v\in(0,1] ϵ u , ϵ v ∈ ( 0 , 1 ]
A u = ϵ u ( B R ) − a ∗ m , A v = ϵ v ( B R ) − b ∗ m , (69) A_u=\epsilon_u(BR)^{-a_*m},\qquad A_v=\epsilon_v(BR)^{-b_*m}, \tag*{(69)} A u = ϵ u ( B R ) − a ∗ m , A v = ϵ v ( B R ) − b ∗ m , (69) we define
u = A u σ u S ι i n / p , v = A v σ v S ι i n / p (70) u=A_u\sigma_u S\iota_{\mathrm{in}}/p,\qquad v=A_v\sigma_v S\iota_{\mathrm{in}}/p \tag*{(70)} u = A u σ u S ι in / p , v = A v σ v S ι in / p (70) on both blocks of each pair, and u = v = 0 u=v=0 u = v = 0 Q Q Q S S S p p p [62 Section 2, p. 1045] or random signs [22 Section 3.5.2] , [49 Sections 3–4] ; here the signs are correlated with the phase of the density.
Lemma 6.2 (Lattice priors). Let J ≥ 0 J\ge0 J ≥ 0 M ≥ 2 M\ge2 M ≥ 2 m ≤ c ∗ M m\le c_*M m ≤ c ∗ M B B B d d d C , C Γ ≥ 1 C,C_\Gamma\ge1 C , C Γ ≥ 1 C E ≥ 0 C_E\ge0 C E ≥ 0 d , α , β , r ± d,\alpha,\beta,r_\pm d , α , β , r ± n , B , J , M , ϵ u , ϵ v n,B,J,M,\epsilon_u,\epsilon_v n , B , J , M , ϵ u , ϵ v
(a) They are admissible with frequency M M M C 0 = 1 / r − C_0=1/r_- C 0 = 1/ r − p p p C ∞ C^\infty C ∞
r − + δ n ≤ p ≤ r + − δ n , (71) r_-+\delta_n\le p\le r_+-\delta_n, \tag*{(71)} r − + δ n ≤ p ≤ r + − δ n , (71) and the functions u u u v v v Q Q Q
(b) In every realization, u u u v v v C ∞ C^\infty C ∞
∥ u ∥ C α ≤ C ϵ u , ∥ v ∥ C β ≤ C ϵ v , ∥ u ∥ ∞ ≤ A u / r − , ∥ v ∥ ∞ ≤ A v / r − . (72) \lVert u\rVert_{C^\alpha}\le C\epsilon_u,\qquad\lVert v\rVert_{C^\beta}\le C\epsilon_v,\qquad\lVert u\rVert_\infty\le A_u/r_-,\qquad\lVert v\rVert_\infty\le A_v/r_-. \tag*{(72)} ∥ u ∥ C α ≤ C ϵ u , ∥ v ∥ C β ≤ C ϵ v , ∥ u ∥ ∞ ≤ A u / r − , ∥ v ∥ ∞ ≤ A v / r − . (72) Let K K K N N N ( 0 , 0 ) (0,0) ( 0 , 0 ) A u + A v A_u+A_v A u + A v K K K
K ( 0 , v ) = 0 on N ⟹ ∥ K ( u , v ) ∥ C α ≤ C K ϵ u , K(0,v)=0\text{ on }N\Longrightarrow\lVert K(u,v)\rVert_{C^\alpha}\le C_K\epsilon_u, K ( 0 , v ) = 0 on N ⟹ ∥ K ( u , v ) ∥ C α ≤ C K ϵ u , K ( u , 0 ) = 0 on N ⟹ ∥ K ( u , v ) ∥ C β ≤ C K ϵ v , (73) K(u,0)=0\text{ on }N\Longrightarrow\lVert K(u,v)\rVert_{C^\beta}\le C_K\epsilon_v, \tag*{(73)} K ( u , 0 ) = 0 on N ⟹ ∥ K ( u , v ) ∥ C β ≤ C K ϵ v , (73) where C K C_K C K K K K C C C
(c) For every pair and every j ≥ 0 j\ge0 j ≥ 0 Γ j \Gamma_j Γ j Lemma 5.2 satisfy (57) , that is,
∥ Γ j ∥ 2 2 ≤ C Γ j + 1 A u 2 A v 2 ϵ C E M R 1 − M + ( j − M ) M 1 j ≥ M , \lVert\Gamma_j\rVert_2^2\le C_\Gamma^{j+1}A_u^2A_v^2\epsilon^{C_E}M R^{1-M+(j-M)}\sqrt{M}\mathbf{1}_{j\ge M}, ∥ Γ j ∥ 2 2 ≤ C Γ j + 1 A u 2 A v 2 ϵ C E M R 1 − M + ( j − M ) M 1 j ≥ M , where the norm is that of L 2 ( ( U × { 1 , 2 } ) j + 2 ) L^2((\mathcal{U}\times\{1,2\})^{j+2}) L 2 (( U × { 1 , 2 } ) j + 2 )
(d) With ρ n \rho_n ρ n
( E + − E − ) ∫ [ 0 , 1 ] d p u v ≥ c 2 A u A v ρ n M , c 2 = 9 v 0 I i n 16 r 0 . (E_+ - E_-)\int_{[0,1]^d} p_{uv} \ge c_2 A_u A_v \rho_n^M,\qquad c_2 = \frac{9v_0 I_{\mathrm{in}}}{16r_0}. ( E + − E − ) ∫ [ 0 , 1 ] d p uv ≥ c 2 A u A v ρ n M , c 2 = 16 r 0 9 v 0 I in . Block-constant phases. For J = 0 J = 0 J = 0 ϕ = 0 \phi= 0 ϕ = 0 S = 1 S = 1 S = 1 R = m = 1 R = m = 1 R = m = 1 U ∘ = U \mathcal{U}^{\circ} = \mathcal{U} U ∘ = U Θ \Theta Θ p = r 0 ± h n cos Θ p = r_0 \pm h_n \cos\Theta p = r 0 ± h n cos Θ p u = A u σ u ι i n p_u = A_u \sigma_u \iota_{\mathrm{in}} p u = A u σ u ι in p v = A v σ v ι i n p_v = A_v \sigma_v \iota_{\mathrm{in}} p v = A v σ v ι in Θ \Theta Θ M M M 1 / p 1/p 1/ p ρ n M \rho_n^M ρ n M 1 ≤ c ∗ M 1 \le c_*M 1 ≤ c ∗ M n n n P \mathcal{P} P T T T c 1 > 0 c_1 > 0 c 1 > 0 n n n
inf T ^ ∈ P P { ∣ T ^ − T ( P ) ∣ ≥ c 1 m 2 a n } ≥ 3 8 (74) \inf_{\widehat{T}\in\mathcal{P}} \mathbb{P}\{|\widehat{T} - T(P)| \ge c_1m^2a_n\} \ge\frac{3}{8} \tag*{(74)} T ∈ P inf P { ∣ T − T ( P ) ∣ ≥ c 1 m 2 a n } ≥ 8 3 (74) with m = 1 m = 1 m = 1 J J J m ≤ c ∗ M m \le c_*M m ≤ c ∗ M m 2 ≍ log n m^2 \asymp\log n m 2 ≍ log n
Proof of the lemma on lattice priors
We use the Hölder norm ∥ ⋅ ∥ C t \|\cdot\|_{C^t} ∥ ⋅ ∥ C t [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d ∥ ⋅ ∥ C t ( U ) \|\cdot\|_{C^t(\mathcal{U})} ∥ ⋅ ∥ C t ( U ) U \mathcal{U} U
Proof of Lemma 6.2(a). We showed above that p p p C ∞ C^\infty C ∞ p 0 p_0 p 0 Q \mathcal{Q} Q u u u v v v Q \mathcal{Q} Q p ≥ r − p \ge r_- p ≥ r − 0 ≤ S , ι i n ≤ 1 0 \le S,\iota_{\mathrm{in}} \le1 0 ≤ S , ι in ≤ 1 ∣ u ∣ ≤ A u / r − |u| \le A_u/r_- ∣ u ∣ ≤ A u / r − ∣ v ∣ ≤ A v / r − |v| \le A_v/r_- ∣ v ∣ ≤ A v / r − C 0 = 1 / r − C_0 = 1/r_- C 0 = 1/ r − H = ξ H = \xi H = ξ p u = A u σ u S ι i n p_u = A_u\sigma_u S\iota_{\mathrm{in}} p u = A u σ u S ι in p v = A v σ v S ι i n p_v = A_v\sigma_v S\iota_{\mathrm{in}} p v = A v σ v S ι in ℓ \ell ℓ
p c = p 0 + ι o u t ( r 0 − p 0 ) , p 1 = ( − 1 ) ℓ − 1 h n ι o u t cos ϕ , p 2 = − ( − 1 ) ℓ − 1 h n ι o u t sin ϕ , p_c = p_0 + \iota_{\mathrm{out}}(r_0-p_0),\qquad
p_1 = (-1)^{\ell-1}h_n\iota_{\mathrm{out}}\cos\phi,\qquad
p_2 = -(-1)^{\ell-1}h_n\iota_{\mathrm{out}}\sin\phi, p c = p 0 + ι out ( r 0 − p 0 ) , p 1 = ( − 1 ) ℓ − 1 h n ι out cos ϕ , p 2 = − ( − 1 ) ℓ − 1 h n ι out sin ϕ , and h u = h v = S ι i n h_u = h_v = S\iota_{\mathrm{in}} h u = h v = S ι in ξ \xi ξ Θ \Theta Θ
Proof of Lemma 6.2(b). We work in block coordinates on a block with label ℓ \ell ℓ k max = ⌈ max ( α , β ) ⌉ = Q − 2 k_{\max} = \lceil\max(\alpha,\beta)\rceil= Q - 2 k m a x = ⌈ max ( α , β )⌉ = Q − 2
f ( y ) = 1 r 0 + ( − 1 ) ℓ − 1 h n cos ( Θ + y ) , H = S ι i n f ( ϕ ) . f(y) = \frac{1}{r_0 + (-1)^{\ell-1}h_n\cos(\Theta+y)},\qquad H = S\iota_{\mathrm{in}}f(\phi). f ( y ) = r 0 + ( − 1 ) ℓ − 1 h n cos ( Θ + y ) 1 , H = S ι in f ( ϕ ) . Since p = r p = r p = r ι i n \iota_{\mathrm{in}} ι in H = S ι i n / p H = S\iota_{\mathrm{in}}/p H = S ι in / p ∣ H ∣ ≤ 1 / r − |H| \le1/r_- ∣ H ∣ ≤ 1/ r − f f f k max + 1 k_{\max}+1 k m a x + 1 r ± r_{\pm} r ± k max k_{\max} k m a x [ r − , r + ] [r_-,r_+] [ r − , r + ] d , α , β , r ± , ι i n , Υ d,\alpha,\beta,r_\pm,\iota_{\mathrm{in}},\Upsilon d , α , β , r ± , ι in , Υ n , B , J , M , ϵ u , ϵ v n,B,J,M,\epsilon_u,\epsilon_v n , B , J , M , ϵ u , ϵ v
Compositions. Let κ \kappa κ C ∞ C^\infty C ∞ [ − 1 / r − , 1 / r − ] [-1/r_-,1/r_-] [ − 1/ r − , 1/ r − ] κ ( 0 ) = 0 \kappa(0)=0 κ ( 0 ) = 0 s 0 ≤ t ≤ max ( α , β ) s_0\le t\le\max(\alpha,\beta) s 0 ≤ t ≤ max ( α , β )
∥ κ ( H ) ∥ C t ( U ) ≤ C κ ( 1 + 2 J t / m ) , (75) \lVert\kappa(H)\rVert_{C^t(U)}\le C_\kappa(1+2^{Jt}/m), \tag*{(75)} ∥ κ ( H ) ∥ C t ( U ) ≤ C κ ( 1 + 2 J t / m ) , (75) where C κ C_\kappa C κ κ \kappa κ ⌈ t ⌉ + 1 \lceil t\rceil+1 ⌈ t ⌉ + 1
w a = λ a η a = γ a 2 ( J − a ) α 0 m , L a = 2 a γ a , ϕ a = ∑ c = 1 a ∑ q = 1 d ξ c , q s c , q ( 0 ≤ a ≤ J ) . w_a=\lambda_a\eta_a=\frac{\gamma_a2^{(J-a)\alpha_0}}{m},\qquad L_a=\frac{2^a}{\gamma_a},\qquad\phi_a=\sum_{c=1}^{a}\sum_{q=1}^{d}\xi_{c,q}s_{c,q}\quad(0\le a\le J). w a = λ a η a = m γ a 2 ( J − a ) α 0 , L a = γ a 2 a , ϕ a = c = 1 ∑ a q = 1 ∑ d ξ c , q s c , q ( 0 ≤ a ≤ J ) . Since γ a ≤ 1 / 4 \gamma_a\le1/4 γ a ≤ 1/4 2 ( J − a ) α 0 ≤ 2 J s 0 = m 2^{(J-a)\alpha_0}\le2^{Js_0}=m 2 ( J − a ) α 0 ≤ 2 J s 0 = m w a ≤ 1 w_a\le1 w a ≤ 1 L a ≥ 1 L_a\ge1 L a ≥ 1 s a , q s_{a,q} s a , q x q x_q x q ∣ D j s a , q ∣ ≤ C L a ∣ j ∣ \lvert D^{\mathbf j}s_{a,q}\rvert\le C L_a^{\lvert\mathbf j\rvert} ∣ D j s a , q ∣ ≤ C L a ∣ j ∣ j \mathbf j j j = 0 \mathbf j=0 j = 0 1 ≤ k ≤ ⌈ t ⌉ 1\le k\le\lceil t\rceil 1 ≤ k ≤ ⌈ t ⌉ b = J − a b=J-a b = J − a w a − i L a − i k / ( w a L a k ) = { ( b + i + 1 ) / ( b + 1 ) } 2 ( k − 1 ) 2 − i ( k − α 0 ) w_{a-i}L_{a-i}^k/(w_aL_a^k)=\{(b+i+1)/(b+1)\}^{2(k-1)}2^{-i(k-\alpha_0)} w a − i L a − i k / ( w a L a k ) = {( b + i + 1 ) / ( b + 1 ) } 2 ( k − 1 ) 2 − i ( k − α 0 ) 0 ≤ i < a 0\le i<a 0 ≤ i < a
∑ c = 1 a ∑ q = 1 d w c L c k ≤ d w a L a k ∑ i ≥ 0 ( i + 1 ) 2 ( k − 1 ) 2 − i ( k − α 0 ) ≤ C w a L a k , (76) \sum_{c=1}^{a}\sum_{q=1}^{d}w_cL_c^k\le dw_aL_a^k\sum_{i\ge0}(i+1)^{2(k-1)}2^{-i(k-\alpha_0)}\le Cw_aL_a^k, \tag*{(76)} c = 1 ∑ a q = 1 ∑ d w c L c k ≤ d w a L a k i ≥ 0 ∑ ( i + 1 ) 2 ( k − 1 ) 2 − i ( k − α 0 ) ≤ C w a L a k , (76) where the series converges because k ≥ 1 > α 0 k\ge1>\alpha_0 k ≥ 1 > α 0
Let F a = κ ( S ι i n f ( ϕ a ) ) F_a=\kappa(S_{\iota_{\mathrm{in}}}f(\phi_a)) F a = κ ( S ι in f ( ϕ a )) V a = F a − F a − 1 V_a=F_a-F_{a-1} V a = F a − F a − 1 W a = ϕ a − ϕ a − 1 W_a=\phi_a-\phi_{a-1} W a = ϕ a − ϕ a − 1 1 ≤ a ≤ J 1\le a\le J 1 ≤ a ≤ J
V a = S ι i n W a ∫ 0 1 f ′ ( z a , u ) κ ′ ( S ι i n f ( z a , u ) ) d u , z a , u = ϕ a − 1 + u W a , V_a=S_{\iota_{\mathrm{in}}}W_a\int_0^1 f'(z_{a,u})\kappa'(S_{\iota_{\mathrm{in}}}f(z_{a,u}))\,\mathrm du,\qquad z_{a,u}=\phi_{a-1}+uW_a, V a = S ι in W a ∫ 0 1 f ′ ( z a , u ) κ ′ ( S ι in f ( z a , u )) d u , z a , u = ϕ a − 1 + u W a , and the arguments of κ \kappa κ [ 0 , 1 / r − ] [0,1/r_-] [ 0 , 1/ r − ] 0 ≤ ∣ j ∣ ≤ ⌈ t ⌉ 0\le\lvert\mathbf j\rvert\le\lceil t\rceil 0 ≤ ∣ j ∣ ≤ ⌈ t ⌉ D j V a D^{\mathbf j}V_a D j V a ∣ j ∣ \lvert\mathbf j\rvert ∣ j ∣ d d d S ϱ S^\varrho S ϱ ϱ ≥ 1 \varrho\ge1 ϱ ≥ 1 ι i n \iota_{\mathrm{in}} ι in f f f κ \kappa κ ∣ j ∣ + 1 \lvert\mathbf j\rvert+1 ∣ j ∣ + 1 D j 0 W a D^{\mathbf j_0}W_a D j 0 W a D j l z a , u D^{\mathbf j_l}z_{a,u} D j l z a , u 1 ≤ l ≤ r 1\le l\le r 1 ≤ l ≤ r ∣ j l ∣ ≥ 1 \lvert\mathbf j_l\rvert\ge1 ∣ j l ∣ ≥ 1 l ≥ 1 l\ge1 l ≥ 1 ∣ j 0 ∣ + ∑ l ∣ j l ∣ ≤ ∣ j ∣ \lvert\mathbf j_0\rvert+\sum_l\lvert\mathbf j_l\rvert\le\lvert\mathbf j\rvert ∣ j 0 ∣ + ∑ l ∣ j l ∣ ≤ ∣ j ∣ r ≤ ∣ j ∣ r\le\lvert\mathbf j\rvert r ≤ ∣ j ∣ W a W_a W a z a , u z_{a,u} z a , u
ξ a , q 0 D j 0 s a , q 0 ∏ l = 1 r ξ c l , q l D j l s c l , q l ( c l ≤ a ) , \xi_{a,q_0}D^{\mathbf j_0}s_{a,q_0}\prod_{l=1}^{r}\xi_{c_l,q_l}D^{\mathbf j_l}s_{c_l,q_l}\qquad(c_l\le a), ξ a , q 0 D j 0 s a , q 0 l = 1 ∏ r ξ c l , q l D j l s c l , q l ( c l ≤ a ) , with at most ∣ j ∣ + 1 ≤ k max + 1 = Q − 1 ≤ 2 Q − 2 \lvert\mathbf j\rvert+1\le k_{\max}+1=Q-1\le2Q-2 ∣ j ∣ + 1 ≤ k m a x + 1 = Q − 1 ≤ 2 Q − 2 S ϱ ≤ S S^\varrho\le S S ϱ ≤ S S ϱ S^\varrho S ϱ C w a L a ∣ j 0 ∣ ∏ l w c l L c l ∣ j l ∣ Cw_aL_a^{\lvert\mathbf j_0\rvert}\prod_lw_{c_l}L_{c_l}^{\lvert\mathbf j_l\rvert} C w a L a ∣ j 0 ∣ ∏ l w c l L c l ∣ j l ∣ q 0 q_0 q 0 c l ≤ a c_l\le a c l ≤ a q l q_l q l w a ≤ 1 w_a\le1 w a ≤ 1 L a ≥ 1 L_a\ge1 L a ≥ 1
∥ D j V a ∥ ∞ ≤ C κ w a L a ∣ j ∣ ( 0 ≤ ∣ j ∣ ≤ ⌈ t ⌉ ) . \lVert D^{\mathbf j}V_a\rVert_\infty\le C_\kappa w_aL_a^{\lvert\mathbf j\rvert}\qquad(0\le\lvert\mathbf j\rvert\le\lceil t\rceil). ∥ D j V a ∥ ∞ ≤ C κ w a L a ∣ j ∣ ( 0 ≤ ∣ j ∣ ≤ ⌈ t ⌉) . With N = ⌈ t ⌉ − 1 N=\lceil t\rceil-1 N = ⌈ t ⌉ − 1 ϑ = t − N ∈ ( 0 , 1 ] \vartheta=t-N\in(0,1] ϑ = t − N ∈ ( 0 , 1 ] ∣ j ∣ = N \lvert j\rvert=N ∣ j ∣ = N
∣ D j V a ( x ) − D j V a ( y ) ∣ ≤ C κ w a L a N min { 1 , L a ∣ x − y ∣ } ≤ C κ w a L a t ∣ x − y ∣ ϑ \lvert D^{j}V_a(x)-D^{j}V_a(y)\rvert\le C_{\kappa}w_aL_a^N\min\{1,L_a\lvert x-y\rvert\}\le C_{\kappa}w_aL_a^t\lvert x-y\rvert^\vartheta ∣ D j V a ( x ) − D j V a ( y )∣ ≤ C κ w a L a N min { 1 , L a ∣ x − y ∣} ≤ C κ w a L a t ∣ x − y ∣ ϑ where we used min ( 1 , z ) ≤ z ϑ \min(1,z)\le z^\vartheta min ( 1 , z ) ≤ z ϑ z ≥ 0 z\ge0 z ≥ 0 L a ≥ 1 L_a\ge1 L a ≥ 1 C κ w a L a t C_{\kappa}w_aL_a^t C κ w a L a t ∥ V a ∥ C t ( U ) ≤ C κ w a L a t \lVert V_a\rVert_{C^t(U)}\le C_{\kappa}w_aL_a^t ∥ V a ∥ C t ( U ) ≤ C κ w a L a t F 0 = κ ( S ι i n f ( 0 ) ) F_0=\kappa(S\iota_{\mathrm{in}}f(0)) F 0 = κ ( S ι in f ( 0 )) C κ C_\kappa C κ S f ( 0 ) ∈ [ 0 , 1 / r − ] Sf(0)\in[0,1/r_-] S f ( 0 ) ∈ [ 0 , 1/ r − ] κ ( H ) = F J = F 0 + ∑ a = 1 J V a \kappa(H)=F_J=F_0+\sum_{a=1}^J V_a κ ( H ) = F J = F 0 + ∑ a = 1 J V a
∥ κ ( H ) ∥ C t ( U ) ≤ C κ { 1 + 2 J t m γ ∗ 1 − t ∑ b ≥ 0 ( b + 1 ) 2 ( t − 1 ) 2 − b ( t − α 0 ) } ≤ C κ ( 1 + 2 J t / m ) , \lVert\kappa(H)\rVert_{C^t(U)}\le C_\kappa\left\{1+\frac{2^{Jt}}{m}\gamma_*^{1-t}\sum_{b\ge0}(b+1)^{2(t-1)}2^{-b(t-\alpha_0)}\right\}\le C_\kappa(1+2^{Jt}/m), ∥ κ ( H ) ∥ C t ( U ) ≤ C κ { 1 + m 2 J t γ ∗ 1 − t b ≥ 0 ∑ ( b + 1 ) 2 ( t − 1 ) 2 − b ( t − α 0 ) } ≤ C κ ( 1 + 2 J t / m ) , where the series is bounded uniformly in s 0 ≤ t ≤ max ( α , β ) s_0\le t\le\max(\alpha,\beta) s 0 ≤ t ≤ max ( α , β ) s 0 > α 0 s_0>\alpha_0 s 0 > α 0 J = 0 J=0 J = 0 a a a
Conclusion. On a block, we have u = A u σ u H u=A_u\sigma_uH u = A u σ u H v = A v σ v H v=A_v\sigma_vH v = A v σ v H g g g C ∞ C^\infty C ∞ [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d Q Q Q g k g_k g k g k g_k g k ∂ U \partial U ∂ U ℓ B = ( v 0 / B ) 1 / d ≤ 1 \ell_B=(v_0/B)^{1/d}\le1 ℓ B = ( v 0 / B ) 1/ d ≤ 1 ∣ j ∣ ≤ t \lvert j\rvert\le t ∣ j ∣ ≤ t ℓ B − ∣ j ∣ ≤ ℓ B − t \ell_B^{-\lvert j\rvert}\le\ell_B^{-t} ℓ B − ∣ j ∣ ≤ ℓ B − t ϑ \vartheta ϑ ⌈ t ⌉ − 1 \lceil t\rceil-1 ⌈ t ⌉ − 1 ℓ B − t \ell_B^{-t} ℓ B − t x x x y y y x x x y y y x x x x ′ x' x ′ y y y y ′ y' y ′ g g g Q Q Q ϑ ≤ 1 \vartheta\le1 ϑ ≤ 1 x x x y y y
∥ g ∥ C t ( [ 0 , 1 ] d ) ≤ 3 ( B / v 0 ) t / d max k ∥ g k ∥ C t ( U ) . (77) \lVert g\rVert_{C^t([0,1]^d)}\le3(B/v_0)^{t/d}\max_k\lVert g_k\rVert_{C^t(U)}. \tag*{(77)} ∥ g ∥ C t ([ 0 , 1 ] d ) ≤ 3 ( B / v 0 ) t / d k max ∥ g k ∥ C t ( U ) . (77) We apply (6.14) with κ ( z ) = z \kappa(z)=z κ ( z ) = z t = α t=\alpha t = α 2 J α = R a ∗ 2^{J\alpha}=R^{a_*} 2 J α = R a ∗ A u B a ∗ = ϵ u R − a ∗ m A_uB^{a_*}=\epsilon_uR^{-a_*}m A u B a ∗ = ϵ u R − a ∗ m m = 2 J s 0 ≤ R a ∗ m=2^{Js_0}\le R^{a_*} m = 2 J s 0 ≤ R a ∗
∥ u ∥ C α ≤ C A u B a ∗ ( 1 + R a ∗ / m ) = C ϵ u ( m R − a ∗ + 1 ) ≤ 2 C ϵ u . \lVert u\rVert_{C^\alpha}\le CA_uB^{a_*}(1+R^{a_*}/m)=C\epsilon_u(mR^{-a_*}+1)\le2C\epsilon_u. ∥ u ∥ C α ≤ C A u B a ∗ ( 1 + R a ∗ / m ) = C ϵ u ( m R − a ∗ + 1 ) ≤ 2 C ϵ u . The bound for v v v β \beta β b ∗ b_* b ∗ ϵ v \epsilon_v ϵ v
Let K K K ( 0 , 0 ) (0,0) ( 0 , 0 ) K ( 0 , v ) = 0 K(0,v)=0 K ( 0 , v ) = 0 N ′ N' N ′ ( 0 , 0 ) (0,0) ( 0 , 0 ) N ′ N' N ′ K ( u , v ) = u K ~ ( u , v ) K(u,v)=u\widetilde K(u,v) K ( u , v ) = u K ( u , v ) K ~ ( u , v ) = ∫ 0 1 ∂ u K ( s u , v ) d s \widetilde K(u,v)=\int_0^1\partial_uK(su,v)\,ds K ( u , v ) = ∫ 0 1 ∂ u K ( s u , v ) d s ( A u + A v ) / r − (A_u+A_v)/r_- ( A u + A v ) / r − N ′ N' N ′ K ( u , v ) = A u σ u κ ( H ) K(u,v)=A_u\sigma_u\kappa(H) K ( u , v ) = A u σ u κ ( H ) κ ( z ) = z K ~ ( A u σ u z , A v σ v z ) \kappa(z)=z\widetilde K(A_u\sigma_uz,A_v\sigma_vz) κ ( z ) = z K ( A u σ u z , A v σ v z ) K ( u , v ) = 0 K(u,v)=0 K ( u , v ) = 0 κ \kappa κ k max + 1 k_{\max}+1 k m a x + 1 [ − 1 / r − , 1 / r − ] [-1/r_-,1/r_-] [ − 1/ r − , 1/ r − ] K K K N ′ N' N ′ A u ≤ ϵ u B − a ∗ ≤ 1 A_u\le\epsilon_uB^{-a_*}\le1 A u ≤ ϵ u B − a ∗ ≤ 1 A v ≤ ϵ v B − b ∗ ≤ 1 A_v\le\epsilon_vB^{-b_*}\le1 A v ≤ ϵ v B − b ∗ ≤ 1 κ \kappa κ t = α t=\alpha t = α ∥ K ( u , v ) ∥ C α ≤ C K ϵ u \lVert K(u,v)\rVert_{C^\alpha}\le C_K\epsilon_u ∥ K ( u , v ) ∥ C α ≤ C K ϵ u K ( u , 0 ) = 0 K(u,0)=0 K ( u , 0 ) = 0 β \beta β ϵ v \epsilon_v ϵ v
Proof of Lemma 6.2(c). We fix a pair. Since p u = A u σ u S i n p_u=A_{u}\sigma_uS_{\mathrm{in}} p u = A u σ u S in p v = A v σ v S i n p_v=A_{v}\sigma_vS_{\mathrm{in}} p v = A v σ v S in
Γ j ( x , y , ζ 1 , … , ζ j ) = 2 A u A v ι i n ( x ) ι i n ( y ) E [ S 2 cos ( M Θ ) ∏ i = 1 j { p ( ζ i ) − p ˉ } ] . \Gamma_j(x,y,\zeta_1,\ldots,\zeta_j)=2A_uA_v\iota_{\mathrm{in}}(x)\iota_{\mathrm{in}}(y)\mathbb{E}\left[S^2\cos(M\Theta)\prod_{i=1}^{j}\{p(\zeta_i)-\bar p\}\right]. Γ j ( x , y , ζ 1 , … , ζ j ) = 2 A u A v ι in ( x ) ι in ( y ) E [ S 2 cos ( M Θ ) i = 1 ∏ j { p ( ζ i ) − p ˉ } ] . Since p p p p ˉ \bar p p ˉ [ r − , r + ] [r_-,r_+] [ r − , r + ] ∣ p − p ˉ ∣ ≤ r + |p-\bar p|\le r_+ ∣ p − p ˉ ∣ ≤ r + ∥ Γ j ∥ ∞ ≤ 2 A u A v r + j \|\Gamma_j\|_\infty\le2A_uA_vr_+^j ∥ Γ j ∥ ∞ ≤ 2 A u A v r + j ∥ Γ j ∥ 2 2 ≤ ∥ Γ j ∥ ∞ ∥ Γ j ∥ 1 \|\Gamma_j\|_2^2\le\|\Gamma_j\|_\infty\|\Gamma_j\|_1 ∥ Γ j ∥ 2 2 ≤ ∥ Γ j ∥ ∞ ∥ Γ j ∥ 1
∥ Γ j ∥ 1 ≤ 8 A u A v ( 6 e r + ) j e C E M R 1 − M + ( j − M ) M 1 j ≥ M , (78) \|\Gamma_j\|_1\le8A_uA_v(6er_+)^j e^{C_E M}R^{1-M+(j-M)\sqrt{M}}\mathbf{1}_{j\ge M}, \tag*{(78)} ∥ Γ j ∥ 1 ≤ 8 A u A v ( 6 e r + ) j e C E M R 1 − M + ( j − M ) M 1 j ≥ M , (78) where e = exp ( 1 ) e=\exp(1) e = exp ( 1 ) C Γ = 16 ( 1 + 6 e r + 2 ) C_\Gamma=16(1+6er_+^2) C Γ = 16 ( 1 + 6 e r + 2 ) ( U × { 1 , 2 } ) j + 2 (\mathcal{U}\times\{1,2\})^{j+2} ( U × { 1 , 2 } ) j + 2 2 j + 2 2^{j+2} 2 j + 2
∥ Γ j ∥ 1 ≤ 8 A u A v ( 2 r + ) j . (79) \|\Gamma_j\|_1\le8A_uA_v(2r_+)^j. \tag*{(79)} ∥ Γ j ∥ 1 ≤ 8 A u A v ( 2 r + ) j . (79) If j ≥ M + M j\ge M+\sqrt{M} j ≥ M + M ( j − M ) M ≥ M (j-M)\sqrt{M}\ge M ( j − M ) M ≥ M
We expand each factor in Fourier modes in Θ \Theta Θ ℓ \ell ℓ
p ( ζ ) − p ˉ = c ( ζ ) + w ( ζ ) ∑ o = ± 1 e i o ( Θ + ϕ ( ζ ) ) , p(\zeta)-\bar p=c(\zeta)+w(\zeta)\sum_{o=\pm1}e^{io(\Theta+\phi(\zeta))}, p ( ζ ) − p ˉ = c ( ζ ) + w ( ζ ) o = ± 1 ∑ e i o ( Θ + ϕ ( ζ )) , c = ( ι o u t − ι ˉ ) ( r 0 − p 0 ) , w = ( − 1 ) ℓ − 1 ι o u t h n / 2 , c=(\iota_{\mathrm{out}}-\bar\iota)(r_0-p_0),\qquad w=(-1)^{\ell-1}\iota_{\mathrm{out}}h_n/2, c = ( ι out − ι ˉ ) ( r 0 − p 0 ) , w = ( − 1 ) ℓ − 1 ι out h n /2 , and ∣ c ∣ ≤ r + |c|\le r_+ ∣ c ∣ ≤ r + ∣ w ∣ ≤ r + / 2 |w|\le r_+/2 ∣ w ∣ ≤ r + /2 cos ( M Θ ) = 1 2 ∑ ε = ± 1 e i ε M Θ \cos(M\Theta)=\frac{1}{2}\sum_{\varepsilon=\pm1}e^{i\varepsilon M\Theta} cos ( M Θ ) = 2 1 ∑ ε = ± 1 e i εM Θ ε \varepsilon ε o i ∈ { − 1 , 0 , 1 } o_i\in\{-1,0,1\} o i ∈ { − 1 , 0 , 1 } i ∈ { 1 , … , j } i\in\{1,\ldots,j\} i ∈ { 1 , … , j } Γ j \Gamma_j Γ j F ε , o F_{\varepsilon,o} F ε , o A u A v ι i n ( x ) ι i n ( y ) ∏ o i = 0 c ( ζ i ) ∏ o i ≠ 0 w ( ζ i ) A_uA_v\iota_{\mathrm{in}}(x)\iota_{\mathrm{in}}(y)\prod_{o_i=0}c(\zeta_i)\prod_{o_i\ne0}w(\zeta_i) A u A v ι in ( x ) ι in ( y ) ∏ o i = 0 c ( ζ i ) ∏ o i = 0 w ( ζ i ) S 2 e i ε M Θ ∏ o i ≠ 0 e i o i ( Θ + ϕ ( ζ i ) ) S^2e^{i\varepsilon M\Theta}\prod_{o_i\ne0}e^{io_i(\Theta+\phi(\zeta_i))} S 2 e i εM Θ ∏ o i = 0 e i o i ( Θ + ϕ ( ζ i )) S 2 e i ∑ i o i ϕ ( ζ i ) = ∏ a , q G a ( ξ a , q ) 2 e i ξ a , q Φ a , q S^2e^{i\sum_i o_i\phi(\zeta_i)}=\prod_{a,q}G_a(\xi_{a,q})^2e^{i\xi_{a,q}\Phi_{a,q}} S 2 e i ∑ i o i ϕ ( ζ i ) = ∏ a , q G a ( ξ a , q ) 2 e i ξ a , q Φ a , q
1 { ∑ i o i = − ε M } ∏ a , q Ξ ^ a ( Φ a , q ) , Φ a , q = ∑ i : o i ≠ 0 o i s a , q ( ζ i ) . (80) \mathbf{1}\left\{\sum_i o_i=-\varepsilon M\right\}\prod_{a,q}\widehat{\Xi}_a(\Phi_{a,q}),\qquad\Phi_{a,q}=\sum_{i:o_i\ne0}o_i s_{a,q}(\zeta_i). \tag*{(80)} 1 { i ∑ o i = − εM } a , q ∏ Ξ a ( Φ a , q ) , Φ a , q = i : o i = 0 ∑ o i s a , q ( ζ i ) . (80) Since ∣ ∑ i o i ∣ ≤ j \left|\sum_i o_i\right|\le j ∣ ∑ i o i ∣ ≤ j Γ j = 0 \Gamma_j=0 Γ j = 0 j < M j<M j < M
Now let M ≤ j < M + M M\le j<M+\sqrt{M} M ≤ j < M + M 2 ⋅ 3 j 2\cdot3^j 2 ⋅ 3 j F ε , o F_{\varepsilon,o} F ε , o
∥ F ε , o ∥ 1 ≤ 4 A u A v ( 2 e r + ) j R 1 − M e C E M , (81) \|F_{\varepsilon,o}\|_1\le4A_uA_v(2er_+)^jR^{1-M}e^{C_EM}, \tag*{(81)} ∥ F ε , o ∥ 1 ≤ 4 A u A v ( 2 e r + ) j R 1 − M e C E M , (81) which we prove for a fixed pattern with ∑ i o i = − ε M \sum_i o_i=-\varepsilon M ∑ i o i = − εM I − = { i : o i = − ε } I_-=\{i:o_i=-\varepsilon\} I − = { i : o i = − ε } I + = { i : o i = ε } I_+=\{i:o_i=\varepsilon\} I + = { i : o i = ε } k = ∣ I + ∣ k=|I_+| k = ∣ I + ∣ ∣ I − ∣ = M + k |I_-|=M+k ∣ I − ∣ = M + k ∣ I − ∣ + ∣ I + ∣ ≤ j |I_-|+|I_+|\le j ∣ I − ∣ + ∣ I + ∣ ≤ j 2 k ≤ j − M < M ≤ M 2k\le j-M<\sqrt{M}\le M 2 k ≤ j − M < M ≤ M e ∗ = ( e a , q ∗ ) ∈ { 0 , 1 } J × d e^*=(e^*_{a,q})\in\{0,1\}^{J\times d} e ∗ = ( e a , q ∗ ) ∈ { 0 , 1 } J × d i ∈ I − ∪ I + i\in I_-\cup I_+ i ∈ I − ∪ I + t i , a , q = e a , q ∗ t_{i,a,q}=e^*_{a,q} t i , a , q = e a , q ∗ i ∈ I − i\in I_- i ∈ I − t i , a , q = 1 − e a , q ∗ t_{i,a,q}=1-e^*_{a,q} t i , a , q = 1 − e a , q ∗ i ∈ I + i\in I_+ i ∈ I + χ i , a , q = ∣ s a , q ( ζ i ) − t i , a , q ∣ \chi_{i,a,q}=|s_{a,q}(\zeta_i)-t_{i,a,q}| χ i , a , q = ∣ s a , q ( ζ i ) − t i , a , q ∣ e ∗ e^* e ∗
∑ i ∈ I − ∪ I + χ i , a , q ≤ k + K a for all a , q . (82) \sum_{i\in I_-\cup I_+}\chi_{i,a,q}\le k+K_a\qquad\text{for all }a,q. \tag*{(82)} i ∈ I − ∪ I + ∑ χ i , a , q ≤ k + K a for all a , q . (82) Indeed, we fix a a a q q q X = − ε Φ a , q = ∑ I − s a , q ( ζ i ) − ∑ I + s a , q ( ζ i ) X=-\varepsilon\Phi_{a,q}=\sum_{I_-}s_{a,q}(\zeta_i)-\sum_{I_+}s_{a,q}(\zeta_i) X = − ε Φ a , q = ∑ I − s a , q ( ζ i ) − ∑ I + s a , q ( ζ i ) 0 ≤ s a , q ≤ 1 0\le s_{a,q}\le1 0 ≤ s a , q ≤ 1 − k ≤ X ≤ M + k -k\le X\le M+k − k ≤ X ≤ M + k − M / 2 < X < 3 M / 2 -M/2<X<3M/2 − M /2 < X < 3 M /2 Ξ ^ a ( Φ a , q ) ≠ 0 \widehat{\Xi}_a(\Phi_{a,q})\ne0 Ξ a ( Φ a , q ) = 0 l ′ l' l ′ ∣ Φ a , q − l ′ M ∣ ≤ K a |\Phi_{a,q}-l'M|\le K_a ∣ Φ a , q − l ′ M ∣ ≤ K a l = − ε l ′ l=-\varepsilon l' l = − ε l ′ ∣ X − l M ∣ ≤ K a ≤ M / 4 |X-lM|\le K_a\le M/4 ∣ X − l M ∣ ≤ K a ≤ M /4 (65) , and hence l ∈ { 0 , 1 } l\in\{0,1\} l ∈ { 0 , 1 } e a , q ∗ = l e^*_{a,q}=l e a , q ∗ = l ∣ I − ∣ = M + k |I_-|=M+k ∣ I − ∣ = M + k ∣ I + ∣ = k |I_+|=k ∣ I + ∣ = k (82) equals X + k X+k X + k l = 0 l=0 l = 0 M + k − X M+k-X M + k − X l = 1 l=1 l = 1 k + K a k+K_a k + K a (64) , we also have χ i , a , q ≥ 1 2 1 { e a ( ζ i , q ) ≠ t i , a , q } \chi_{i,a,q}\ge\frac{1}{2}\mathbf{1}\{e_a(\zeta_{i,q})\ne t_{i,a,q}\} χ i , a , q ≥ 2 1 1 { e a ( ζ i , q ) = t i , a , q } ζ i , q \zeta_{i,q} ζ i , q q q q ζ i \zeta_i ζ i
Since the product in (80) has modulus at most one, its modulus is at most the sum, over the R R R e ∗ e^* e ∗ d J dJ dJ (82) hold. We define Ω a = 2 log d + 4 log ( J − a + 2 ) \Omega_a=2\log d+4\log(J-a+2) Ω a = 2 log d + 4 log ( J − a + 2 ) 1 ≤ a ≤ J 1\le a\le J 1 ≤ a ≤ J Ω = d ∑ a Ω a \Omega=d\sum_a\Omega_a Ω = d ∑ a Ω a X ′ ≤ c ′ X'\le c' X ′ ≤ c ′ e Ω a ( c ′ − X ′ ) e^{\Omega_a(c'-X')} e Ω a ( c ′ − X ′ ) a a a q q q e ∗ e^* e ∗
e Ω k exp { d ∑ a Ω a K a } ∏ i ∈ I − ∪ I + exp { − ∑ a , q Ω a χ i , a , q } . e^{\Omega k}\exp\left\{d\sum_a\Omega_aK_a\right\}\prod_{i\in I_-\cup I_+}\exp\left\{-\sum_{a,q}\Omega_a\chi_{i,a,q}\right\}. e Ω k exp { d a ∑ Ω a K a } i ∈ I − ∪ I + ∏ exp { − a , q ∑ Ω a χ i , a , q } . With b = J − a b=J-a b = J − a Ω a K a = { 2 log d + 4 log ( b + 2 ) } K ( b ) m \Omega_aK_a=\{2\log d+4\log(b+2)\}K(b)m Ω a K a = { 2 log d + 4 log ( b + 2 )} K ( b ) m K ( b ) K(b) K ( b ) b b b d ∑ a Ω a K a ≤ C g a t e m d\sum_a\Omega_aK_a\le C_{\mathrm{gate}}m d ∑ a Ω a K a ≤ C gate m C g a t e = d ∑ b ≥ 0 { 2 log d + 4 log ( b + 2 ) } K ( b ) < ∞ C_{\mathrm{gate}}=d\sum_{b\ge0}\{2\log d+4\log(b+2)\}K(b)<\infty C gate = d ∑ b ≥ 0 { 2 log d + 4 log ( b + 2 )} K ( b ) < ∞ Ω ≤ d J { 2 log d + 4 log ( J + 1 ) } \Omega\le dJ\{2\log d+4\log(J+1)\} Ω ≤ dJ { 2 log d + 4 log ( J + 1 )} m = 2 − J s 0 / 2 m=2^{-Js_0/2} m = 2 − J s 0 /2 Ω ≤ D m \Omega\le D\sqrt{m} Ω ≤ D m D = sup J ′ ≥ 1 d J ′ { 2 log d + 4 log ( J ′ + 1 ) } 2 − J ′ s 0 / 2 < ∞ D=\sup_{J'\ge1}dJ'\{2\log d+4\log(J'+1)\}2^{-J's_0/2}<\infty D = sup J ′ ≥ 1 d J ′ { 2 log d + 4 log ( J ′ + 1 )} 2 − J ′ s 0 /2 < ∞ K ≥ K ( 0 ) = 6 Q λ ∗ / γ ∗ > 1 K\ge K(0)=6Q\lambda_*/\gamma_*>1 K ≥ K ( 0 ) = 6 Q λ ∗ / γ ∗ > 1 c ∗ < 1 c_*<1 c ∗ < 1 m ≤ M m\le M m ≤ M 2 k < M 2k<\sqrt{M} 2 k < M Ω k + d ∑ a Ω a K a ≤ C E M \Omega k+d\sum_a\Omega_aK_a\le C_EM Ω k + d ∑ a Ω a K a ≤ C E M C E = C g a t e + D C_E=C_{\mathrm{gate}}+D C E = C gate + D i ∈ I − ∪ I + i\in I_-\cup I_+ i ∈ I − ∪ I + χ i , a , q \chi_{i,a,q} χ i , a , q d e − Ω a / 2 = ( b + 2 ) − 2 de^{-\Omega_a/2}=(b+2)^{-2} d e − Ω a /2 = ( b + 2 ) − 2
∫ U × { 1 , 2 } ∣ w ( ζ i ) ∣ e − ∑ a , q Ω a χ i , a , q d ζ i ≤ r + ∏ a , q 1 + e − Ω a / 2 2 ≤ r + R exp { ∑ b ≥ 0 1 ( b + 2 ) 2 } ≤ e r + R . \int_{U\times\{1,2\}}|w(\zeta_i)|e^{-\sum_{a,q}\Omega_a\chi_{i,a,q}}\,d\zeta_i
\le r_+\prod_{a,q}\frac{1+e^{-\Omega_a/2}}{2}
\le\frac{r_+}{R}\exp\left\{\sum_{b\ge0}\frac{1}{(b+2)^2}\right\}
\le\frac{er_+}{R}. ∫ U × { 1 , 2 } ∣ w ( ζ i ) ∣ e − ∑ a , q Ω a χ i , a , q d ζ i ≤ r + a , q ∏ 2 1 + e − Ω a /2 ≤ R r + exp { b ≥ 0 ∑ ( b + 2 ) 2 1 } ≤ R e r + . Each i i i o i = 0 o_i=0 o i = 0 2 r + 2r_+ 2 r + x x x y y y R R R e ∗ e^* e ∗
∥ F ε , o ∥ 1 ≤ 4 A u A v R e C E M ( e r + R ) M + 2 k ( 2 r + ) j − M − 2 k ≤ 4 A u A v ( 2 e r + ) j R 1 − M e C E M , \|F_{\varepsilon,o}\|_1\le4A_uA_vRe^{C_EM}\left(\frac{er_+}{R}\right)^{M+2k}(2r_+)^{j-M-2k}\le4A_uA_v(2er_+)^jR^{1-M}e^{C_EM}, ∥ F ε , o ∥ 1 ≤ 4 A u A v R e C E M ( R e r + ) M + 2 k ( 2 r + ) j − M − 2 k ≤ 4 A u A v ( 2 e r + ) j R 1 − M e C E M , which proves (81) .
□ \square □
Proof of Lemma 6.2(d). Let I = [ r − + δ n , r + − δ n ] I=[r_-+\delta_n,r_+-\delta_n] I = [ r − + δ n , r + − δ n ] ρ I = ρ n \rho_I=\rho_n ρ I = ρ n c I = r 0 c_I=r_0 c I = r 0 d I = h n d_I=h_n d I = h n g ˉ I = r 0 2 − h n 2 ≤ r 0 \bar g_I=\sqrt{r_0^2-h_n^2}\le r_0 g ˉ I = r 0 2 − h n 2 ≤ r 0 ℓ \ell ℓ p u v = A u A v σ u σ v S i n 2 ℓ i n 2 / r p_{uv}=A_uA_v\sigma_u\sigma_vS_{\mathrm{in}}^2\ell^2_{\mathrm{in}}/r p uv = A u A v σ u σ v S in 2 ℓ in 2 / r r = c I + d I cos ( Θ + ϕ + ( ℓ − 1 ) π ) r=c_I+d_I\cos(\Theta+\phi+(\ell-1)\pi) r = c I + d I cos ( Θ + ϕ + ( ℓ − 1 ) π ) (27) converges absolutely, multiplying it by cos ( M Θ ) \cos(M\Theta) cos ( M Θ ) Θ \Theta Θ ζ \zeta ζ M M M E Θ E_\Theta E Θ M M M
E Θ cos ( M Θ ) r = ρ n M cos ( M ϕ ) g ˉ I . E_\Theta\frac{\cos(M\Theta)}{r}=\frac{\rho_n^M\cos(M\phi)}{\bar g_I}. E Θ r cos ( M Θ ) = g ˉ I ρ n M cos ( M ϕ ) . By (51) , and since Θ \Theta Θ ζ \zeta ζ B B B v 0 / B v_0/B v 0 / B
( E + − E − ) ∫ [ 0 , 1 ] d p u v = 2 v 0 A u A v ρ n M g ˉ I E [ S 2 ∫ U ι i n 2 cos ( M ϕ ) ] . (E_+ - E_-)\int_{[0,1]^d} p_{uv} = \frac{2v_0 A_u A_v \rho_n^M}{\bar{g}_I}\mathbb{E}\left[S^2\int_{\mathcal{U}}\iota_{\mathrm{in}}^2\cos(M\phi)\right]. ( E + − E − ) ∫ [ 0 , 1 ] d p uv = g ˉ I 2 v 0 A u A v ρ n M E [ S 2 ∫ U ι in 2 cos ( M ϕ ) ] . By (67) , cos ( M ϕ ) = 1 \cos(M\phi)=1 cos ( M ϕ ) = 1 U ∘ \mathcal{U}^{\circ} U ∘ 0 ≤ ι i n ≤ 1 0\leq\iota_{\mathrm{in}}\leq1 0 ≤ ι in ≤ 1 (68) implies that the integral is at least I i n − 2 ∣ U ∖ U ∘ ∣ ≥ I i n / 2 I_{\mathrm{in}}-2\lvert\mathcal{U}\setminus\mathcal{U}^{\circ}\rvert\geq I_{\mathrm{in}}/2 I in − 2 ∣ U ∖ U ∘ ∣ ≥ I in /2 ( I i n / 2 ) E S 2 ≥ 9 I i n / 32 (I_{\mathrm{in}}/2)\mathbb{E}S^2\geq9I_{\mathrm{in}}/32 ( I in /2 ) E S 2 ≥ 9 I in /32 S S S ϕ \phi ϕ g ˉ I ≤ r 0 \bar{g}_I\leq r_0 g ˉ I ≤ r 0 c 2 = 9 v 0 I i n / ( 16 r 0 ) c_2=9v_0I_{\mathrm{in}}/(16r_0) c 2 = 9 v 0 I in / ( 16 r 0 ) □ \square □
Embedding and proof of the lower bounds
Embedding the priors in the generic model
We perturb π 0 \pi_0 π 0 a , b a,b a , b u / w , v / w u/w,v/w u / w , v / w ( u + v ) / w (u+v)/w ( u + v ) / w [49 Sections 3–4] , where the responses are binary, we use a finite response space whose conditional laws are perturbed through the nuisance functions.
We write a finite-support law for Z Z Z π 0 = ∑ k = 1 m 0 π 0 , k δ z k \pi_0=\sum_{k=1}^{m_0}\pi_{0,k}\delta_{z_k} π 0 = ∑ k = 1 m 0 π 0 , k δ z k π 0 , k > 0 \pi_{0,k}>0 π 0 , k > 0 w 0 , a 0 , b 0 w_0,a_0,b_0 w 0 , a 0 , b 0 w 0 > 0 w_0>0 w 0 > 0 r = ( U − a 0 D , V − b 0 D ) ⊤ r=(U-a_0D,V-b_0D)^\top r = ( U − a 0 D , V − b 0 D ) ⊤ π 0 \pi_0 π 0 s u , s v s_u,s_v s u , s v π 0 \pi_0 π 0 π 0 \pi_0 π 0 E π 0 s u = E π 0 s v = 0 \mathbb{E}_{\pi_0}s_u=\mathbb{E}_{\pi_0}s_v=0 E π 0 s u = E π 0 s v = 0
E π 0 [ r ( s u , s v ) ] = Id , or U = V , s u = s v , E π 0 [ ( U − a 0 D ) s u ] = 1 , (83) \mathbb{E}_{\pi_0}[r(s_u,s_v)]=\operatorname{Id},\qquad\text{or}\qquad U=V,\ s_u=s_v,\ \mathbb{E}_{\pi_0}[(U-a_0D)s_u]=1, \tag*{(83)} E π 0 [ r ( s u , s v )] = Id , or U = V , s u = s v , E π 0 [( U − a 0 D ) s u ] = 1 , (83) where Id \operatorname{Id} Id 2 × 2 2\times2 2 × 2 diagonal case ; in it we set r d = U − a 0 D r_d=U-a_0D r d = U − a 0 D s d s_d s d diagonal score . If the residual covariance Σ = E π 0 [ r r ⊤ ] \Sigma=\mathbb{E}_{\pi_0}[rr^\top] Σ = E π 0 [ r r ⊤ ] ( s u , s v ) ⊤ = Σ − 1 r (s_u,s_v)^\top=\Sigma^{-1}r ( s u , s v ) ⊤ = Σ − 1 r U = V U=V U = V E π 0 r d 2 > 0 \mathbb{E}_{\pi_0}r_d^2>0 E π 0 r d 2 > 0 s d = r d / E π 0 r d 2 s_d=r_d/\mathbb{E}_{\pi_0}r_d^2 s d = r d / E π 0 r d 2
π u , v ( { z k } ) = π 0 , k { 1 + u s u ( z k ) + v s v ( z k ) } . (84) \pi_{u,v}(\{z_k\})=\pi_{0,k}\{1+us_u(z_k)+vs_v(z_k)\}. \tag*{(84)} π u , v ({ z k }) = π 0 , k { 1 + u s u ( z k ) + v s v ( z k )} . (84) These probabilities sum to one because the scores have mean zero, and, since the support is finite, they are at least π 0 , k / 2 \pi_{0,k}/2 π 0 , k /2 ∣ u ∣ + ∣ v ∣ |u|+|v| ∣ u ∣ + ∣ v ∣ q z k 0 = π 0 , k q_{z_k0}=\pi_{0,k} q z k 0 = π 0 , k q z k u = π 0 , k s u ( z k ) q_{z_ku}=\pi_{0,k}s_u(z_k) q z k u = π 0 , k s u ( z k ) q z k v = π 0 , k s v ( z k ) q_{z_kv}=\pi_{0,k}s_v(z_k) q z k v = π 0 , k s v ( z k )
Let ( ℓ a , ℓ b ) = ( u , v ) (\ell_a,\ell_b)=(u,v) ( ℓ a , ℓ b ) = ( u , v ) ( ℓ a , ℓ b ) = ( u + v , u + v ) (\ell_a,\ell_b)=(u+v,u+v) ( ℓ a , ℓ b ) = ( u + v , u + v ) d u = E π 0 [ D s u ] d_u=\mathbb{E}_{\pi_0}[Ds_u] d u = E π 0 [ D s u ] d v = E π 0 [ D s v ] d_v=\mathbb{E}_{\pi_0}[Ds_v] d v = E π 0 [ D s v ] f f f E π u , v f = E π 0 f + u E π 0 [ f s u ] + v E π 0 [ f s v ] \mathbb{E}_{\pi_{u,v}}f=\mathbb{E}_{\pi_0}f+u\mathbb{E}_{\pi_0}[fs_u]+v\mathbb{E}_{\pi_0}[fs_v] E π u , v f = E π 0 f + u E π 0 [ f s u ] + v E π 0 [ f s v ] E π 0 r = 0 \mathbb{E}_{\pi_0}r=0 E π 0 r = 0 E π u , v r = ( ℓ a , ℓ b ) ⊤ \mathbb{E}_{\pi_{u,v}}r=(\ell_a,\ell_b)^\top E π u , v r = ( ℓ a , ℓ b ) ⊤ a 0 = b 0 a_0=b_0 a 0 = b 0 r r r r d r_d r d U = r 1 + a 0 D U=r_1+a_0D U = r 1 + a 0 D V = r 2 + b 0 D V=r_2+b_0D V = r 2 + b 0 D
w ( u , v ) = w 0 + d u u + d v v , E π u , v U = a 0 w ( u , v ) + ℓ a , E π u , v V = b 0 w ( u , v ) + ℓ b , (85) \begin{aligned}
w(u,v)&=w_0+d_uu+d_vv, \qquad\mathbb{E}_{\pi_{u,v}}U=a_0w(u,v)+\ell_a,\\
\mathbb{E}_{\pi_{u,v}}V&=b_0w(u,v)+\ell_b, \tag*{(85)}
\end{aligned} w ( u , v ) E π u , v V = w 0 + d u u + d v v , E π u , v U = a 0 w ( u , v ) + ℓ a , = b 0 w ( u , v ) + ℓ b , (85) where w ( u , v ) = E π u , v D w(u,v)=\mathbb{E}_{\pi_{u,v}}D w ( u , v ) = E π u , v D
a ( u , v ) = a 0 + ℓ a w ( u , v ) , b ( u , v ) = b 0 + ℓ b w ( u , v ) . (86) a(u,v)=a_0+\frac{\ell_a}{w(u,v)},\qquad b(u,v)=b_0+\frac{\ell_b}{w(u,v)}. \tag*{(86)} a ( u , v ) = a 0 + w ( u , v ) ℓ a , b ( u , v ) = b 0 + w ( u , v ) ℓ b . (86) Suppose that X X X p p p x x x π u ( x ) , v ( x ) \pi_{u(x),v(x)} π u ( x ) , v ( x ) m U = E π u , v U m_U = \mathbb{E}_{\pi_{u,v}}U m U = E π u , v U m V = E π u , v V m_V = \mathbb{E}_{\pi_{u,v}}V m V = E π u , v V w = w ( u , v ) w = w(u,v) w = w ( u , v )
m U m V w = a 0 m V + b 0 m U − a 0 b 0 w + ( m U − a 0 w ) ( m V − b 0 w ) w . \frac{m_Um_V}{w}=a_0m_V+b_0m_U-a_0b_0w+\frac{(m_U-a_0w)(m_V-b_0w)}{w}. w m U m V = a 0 m V + b 0 m U − a 0 b 0 w + w ( m U − a 0 w ) ( m V − b 0 w ) . By (4) and (85) , this identity implies that ψ ( P ) = ∫ p F ( u , v ) \psi(P)=\int pF(u,v) ψ ( P ) = ∫ pF ( u , v )
F ( u , v ) = E π u , v W ~ + λ ℓ a ℓ b w ( u , v ) , W ~ = W + λ ( a 0 V + b 0 U − a 0 b 0 D ) . (87) F(u,v)=\mathbb{E}_{\pi_{u,v}}\widetilde{W}+\lambda\frac{\ell_a\ell_b}{w(u,v)},\qquad\widetilde{W}=W+\lambda(a_0V+b_0U-a_0b_0D). \tag*{(87)} F ( u , v ) = E π u , v W + λ w ( u , v ) ℓ a ℓ b , W = W + λ ( a 0 V + b 0 U − a 0 b 0 D ) . (87) The first term is affine in ( u , v ) (u,v) ( u , v ) F F F w ( u , v ) ≥ w 0 / 2 w(u,v)\ge w_0/2 w ( u , v ) ≥ w 0 /2 ℓ a ℓ b \ell_a\ell_b ℓ a ℓ b F u v ( 0 , 0 ) = λ ∂ u ∂ v ( ℓ a ℓ b ) / w 0 F_{uv}(0,0)=\lambda\partial_u\partial_v(\ell_a\ell_b)/w_0 F uv ( 0 , 0 ) = λ ∂ u ∂ v ( ℓ a ℓ b ) / w 0 λ / w 0 \lambda/w_0 λ / w 0 2 λ / w 0 2\lambda/w_0 2 λ / w 0
Proposition 7.1 (Local lower bounds). Fix d , α , β d,\alpha,\beta d , α , β θ = ( α + β ) / d \theta=(\alpha+\beta)/d θ = ( α + β ) / d π 0 \pi_0 π 0 Z Z Z w 0 > 0 w_0>0 w 0 > 0 D , U , V , W D,U,V,W D , U , V , W λ ≠ 0 \lambda\ne0 λ = 0 s u , s v s_u,s_v s u , s v π 0 \pi_0 π 0 α = β \alpha=\beta α = β π u , v \pi_{u,v} π u , v F F F 0 < r − < 1 < r + 0<r_-<1<r_+ 0 < r − < 1 < r + ϵ > 0 \epsilon>0 ϵ > 0 τ \tau τ [ r − , r + ] [r_-,r_+] [ r − , r + ] § 6.2 .
(a) Suppose that θ < 1 / 2 \theta<1/2 θ < 1/2 n n n P P P O = ( X , Z ) O=(X,Z) O = ( X , Z ) X X X C ∞ C^\infty C ∞ p p p Z Z Z X = x X=x X = x π u ( x ) , v ( x ) \pi_{u(x),v(x)} π u ( x ) , v ( x ) ∥ u ∥ ∞ ≤ C n − α / d \lVert u\rVert_\infty\le Cn^{-\alpha/d} ∥ u ∥ ∞ ≤ C n − α / d ∥ v ∥ ∞ ≤ C n − β / d \lVert v\rVert_\infty\le Cn^{-\beta/d} ∥ v ∥ ∞ ≤ C n − β / d ∥ u ∥ ∞ + ∥ v ∥ ∞ = o ( δ n ) \lVert u\rVert_\infty+\lVert v\rVert_\infty=o(\delta_n) ∥ u ∥ ∞ + ∥ v ∥ ∞ = o ( δ n ) K K K N N N ( 0 , 0 ) (0,0) ( 0 , 0 ) n n n K K K
K ( 0 , v ) = 0 on N ⟹ ∥ K ( u , v ) ∥ C α ≤ C K ϵ , K ( u , 0 ) = 0 on N ⟹ ∥ K ( u , v ) ∥ C β ≤ C K ϵ . (88) \begin{aligned}
K(0,v)=0\text{ on }N&\Longrightarrow\lVert K(u,v)\rVert_{C^\alpha}\le C_K\epsilon,\\
K(u,0)=0\text{ on }N&\Longrightarrow\lVert K(u,v)\rVert_{C^\beta}\le C_K\epsilon. \tag*{(88)}
\end{aligned} K ( 0 , v ) = 0 on N K ( u , 0 ) = 0 on N ⟹ ∥ K ( u , v ) ∥ C α ≤ C K ϵ , ⟹ ∥ K ( u , v ) ∥ C β ≤ C K ϵ . (88) where C K C_K C K ϵ \epsilon ϵ n n n P n h \mathcal{P}_n^h P n h ψ ( P ) = ∫ p F ( u , v ) \psi(P)=\int pF(u,v) ψ ( P ) = ∫ pF ( u , v ) P \mathcal{P} P P n h \mathcal{P}_n^h P n h T T T T ( P ) = ψ ( P ) + t ∗ T(P)=\psi(P)+t_* T ( P ) = ψ ( P ) + t ∗ P n h \mathcal{P}_n^h P n h t ∗ t_* t ∗ c > 0 c>0 c > 0 n n n
inf T ^ sup P ∈ P P { ∣ T ^ − T ( P ) ∣ ≥ c a n log n } ≥ 3 8 , r n ( T , P ) ≥ 3 8 c a n log n , \inf_{\widehat{T}}\sup_{P\in\mathcal{P}}\mathbb{P}\{|\widehat{T}-T(P)|\ge ca_n\log n\}\ge\frac{3}{8},\qquad r_n(T,\mathcal{P})\ge\sqrt{\frac{3}{8}}ca_n\log n, T inf P ∈ P sup P { ∣ T − T ( P ) ∣ ≥ c a n log n } ≥ 8 3 , r n ( T , P ) ≥ 8 3 c a n log n , with a n = a n ( θ , τ ) a_n=a_n(\theta,\tau) a n = a n ( θ , τ )
(b) For every u ∗ u_* u ∗ P t = Unif ( [ 0 , 1 ] d ) ⊗ π u ∗ , t P_t=\operatorname{Unif}([0,1]^d)\otimes\pi_{u_*,t} P t = Unif ([ 0 , 1 ] d ) ⊗ π u ∗ , t t t t ψ ( P t ) = F ( u ∗ , t ) \psi(P_t)=F(u_*,t) ψ ( P t ) = F ( u ∗ , t ) u ∗ ≠ 0 u_*\ne0 u ∗ = 0 c > 0 c>0 c > 0 u ∗ u_* u ∗ P \mathcal{P} P P t P_t P t t t t T T T T ( P t ) = ψ ( P t ) + t ∗ T(P_t)=\psi(P_t)+t_* T ( P t ) = ψ ( P t ) + t ∗ t ∗ t_* t ∗ r n ( T , P ) ≥ c n − 1 / 2 r_n(T,\mathcal{P})\ge cn^{-1/2} r n ( T , P ) ≥ c n − 1/2 n n n
The constants may depend on the fixed quantities above and on $\epsilon.
Proof. For (a), we apply Proposition 5.3 to the response law (7.2), which has the form (5.1), with the integrand F F F [ r − , r + ] [r_-,r_+] [ r − , r + ] M M M c ∗ c_* c ∗ § 7.2 , and let s 0 = min ( α , β ) s_0=\min(\alpha,\beta) s 0 = min ( α , β ) n n n c ∗ M ≥ 1 c_*M\geq1 c ∗ M ≥ 1
J = ⌊ log 2 ( c ∗ M ) s 0 ⌋ ≥ 0 , m = 2 J s 0 , R = 2 J d , J=\left\lfloor\frac{\log_2(c_*M)}{s_0}\right\rfloor\geq0,\qquad m=2^{Js_0},\qquad R=2^{Jd}, J = ⌊ s 0 log 2 ( c ∗ M ) ⌋ ≥ 0 , m = 2 J s 0 , R = 2 J d , so that 2 − s 0 c ∗ M < m ≤ c ∗ M 2^{-s_0}c_*M<m\leq c_*M 2 − s 0 c ∗ M < m ≤ c ∗ M s 0 ≤ α , β s_0\leq\alpha,\beta s 0 ≤ α , β m = R s 0 / d ≤ min ( R a ∗ , R b ∗ ) m=R^{s_0/d}\leq\min(R^{a_*},R^{b_*}) m = R s 0 / d ≤ min ( R a ∗ , R b ∗ ) log R = ( d / s 0 ) log m = O ( log M ) \log R=(d/s_0)\log m=O(\log M) log R = ( d / s 0 ) log m = O ( log M ) ϵ u = ϵ v = min ( ϵ , 1 ) \epsilon_u=\epsilon_v=\min(\epsilon,1) ϵ u = ϵ v = min ( ϵ , 1 ) Lemma 6.2 (a), (c), and (d), for all large n n n B ≥ n B\geq n B ≥ n d d d M M M J J J B B B ϵ u \epsilon_u ϵ u ϵ v \epsilon_v ϵ v Proposition 5.3 , with constants that do not depend on n n n B B B Proposition 5.3 , there exists such a B B B A u ≤ ϵ u B − a ∗ ≤ n − α / d A_u\leq\epsilon_uB^{-a_*}\leq n^{-\alpha/d} A u ≤ ϵ u B − a ∗ ≤ n − α / d A v ≤ ϵ v B − b ∗ ≤ n − β / d A_v\leq\epsilon_vB^{-b_*}\leq n^{-\beta/d} A v ≤ ϵ v B − b ∗ ≤ n − β / d B B B Lemma 6.2 (a) and (b), every realization satisfies (6.10), ∥ u ∥ ∞ ≤ A u / r − \lVert u\rVert_\infty\leq A_u/r_- ∥ u ∥ ∞ ≤ A u / r − ∥ v ∥ ∞ ≤ A v / r − \lVert v\rVert_\infty\leq A_v/r_- ∥ v ∥ ∞ ≤ A v / r − A u + A v → 0 A_u+A_v\to0 A u + A v → 0 n n n n n n K K K ϵ u , ϵ v ≤ ϵ \epsilon_u,\epsilon_v\leq\epsilon ϵ u , ϵ v ≤ ϵ n − min ( α , β ) / d ( log n ) 2 → 0 n^{-\min(\alpha,\beta)/d}(\log n)^2\to0 n − m i n ( α , β ) / d ( log n ) 2 → 0 ∥ u ∥ ∞ + ∥ v ∥ ∞ = o ( δ n ) \lVert u\rVert_\infty+\lVert v\rVert_\infty=o(\delta_n) ∥ u ∥ ∞ + ∥ v ∥ ∞ = o ( δ n )
By (7.5), ψ ( P ) = ∫ p F ( u , v ) \psi(P)=\int pF(u,v) ψ ( P ) = ∫ pF ( u , v ) Proposition 5.3 to T − t ∗ T-t_* T − t ∗ T T T c 1 m 2 a n c_1m^2a_n c 1 m 2 a n m > 2 − s 0 c ∗ M m>2^{-s_0}c_*M m > 2 − s 0 c ∗ M M ≥ θ ( 1 − 2 θ ) log n / τ − 1 M\geq\sqrt{\theta(1-2\theta)\log n/\tau}-1 M ≥ θ ( 1 − 2 θ ) log n / τ − 1
m 2 ≥ 2 − 2 s 0 − 2 c ∗ 2 θ ( 1 − 2 θ ) τ log n m^2\geq2^{-2s_0-2}c_*^2\frac{\theta(1-2\theta)}{\tau}\log n m 2 ≥ 2 − 2 s 0 − 2 c ∗ 2 τ θ ( 1 − 2 θ ) log n for all large n n n c a n log n c\,a_n\log n c a n log n c = 2 − 2 s 0 − 2 c ∗ 2 θ ( 1 − 2 θ ) c 1 / τ c=2^{-2s_0-2}c_*^2\theta(1-2\theta)c_1/\tau c = 2 − 2 s 0 − 2 c ∗ 2 θ ( 1 − 2 θ ) c 1 / τ J = 0 J=0 J = 0 m = R = 1 m=R=1 m = R = 1 a n a_n a n a n log n a_n\log n a n log n
For (b), the constant perturbations u = u ∗ u=u_* u = u ∗ v = t v=t v = t p = 1 p=1 p = 1 P t P_t P t ψ ( P t ) = F ( u ∗ , t ) \psi(P_t)=F(u_*,t) ψ ( P t ) = F ( u ∗ , t ) φ ( u ) = F v ( u , 0 ) \varphi(u)=F_v(u,0) φ ( u ) = F v ( u , 0 ) φ ′ ( 0 ) = F u v ( 0 , 0 ) ≠ 0 \varphi'(0)=F_{uv}(0,0)\ne0 φ ′ ( 0 ) = F uv ( 0 , 0 ) = 0 φ ( u ∗ ) ≠ 0 \varphi(u_*)\ne0 φ ( u ∗ ) = 0 u ∗ ≠ 0 u_*\ne0 u ∗ = 0 φ ( 0 ) ≠ 0 \varphi(0)\ne0 φ ( 0 ) = 0 φ ( u ) = u { φ ′ ( 0 ) + o ( 1 ) } \varphi(u)=u\{\varphi'(0)+o(1)\} φ ( u ) = u { φ ′ ( 0 ) + o ( 1 )} P ∗ = Unif ( [ 0 , 1 ] d ) ⊗ π 0 P_*=\operatorname{Unif}([0,1]^d)\otimes\pi_0 P ∗ = Unif ([ 0 , 1 ] d ) ⊗ π 0 P t P_t P t 1 + u ∗ s u ( Z ) + t s v ( Z ) 1+u_*s_u(Z)+ts_v(Z) 1 + u ∗ s u ( Z ) + t s v ( Z ) 1 / 2 1/2 1/2 ∣ u ∗ ∣ + ∣ t ∣ |u_*|+|t| ∣ u ∗ ∣ + ∣ t ∣ s v ( Z ) s_v(Z) s v ( Z ) t t t P t ∈ P P_t\in\mathcal{P} P t ∈ P t ↦ T ( P t ) = F ( u ∗ , t ) + t ∗ t\mapsto T(P_t)=F(u_*,t)+t_* t ↦ T ( P t ) = F ( u ∗ , t ) + t ∗ φ ( u ∗ ) ≠ 0 \varphi(u_*)\ne0 φ ( u ∗ ) = 0 t = 0 t=0 t = 0 Lemma 5.1 . □
Localization
Every conditional moment of a hard law is affine in ( u , v ) (u,v) ( u , v ) E P [ f ( Z ) ∣ X ] = E π 0 f + u E π 0 [ f s u ] + v E π 0 [ f s v ] \mathbb{E}_P[f(Z)\mid X]=\mathbb{E}_{\pi_0}f+u\mathbb{E}_{\pi_0}[fs_u]+v\mathbb{E}_{\pi_0}[fs_v] E P [ f ( Z ) ∣ X ] = E π 0 f + u E π 0 [ f s u ] + v E π 0 [ f s v ] K P ( x ) = K ( u ( x ) , v ( x ) ) K_P(x) = K(u(x), v(x)) K P ( x ) = K ( u ( x ) , v ( x )) C ∞ C^\infty C ∞ K K K
Lemma 7.2 (Localization and membership). Adopt the setting of Proposition 7.1. Let K K K C ∞ C^\infty C ∞ ( 0 , 0 ) (0, 0) ( 0 , 0 ) k 0 = K ( 0 , 0 ) k_0 = K(0, 0) k 0 = K ( 0 , 0 ) P P P u , v u,v u , v K P ( x ) = K ( u ( x ) , v ( x ) ) K_P(x) = K(u(x), v(x)) K P ( x ) = K ( u ( x ) , v ( x )) γ K = α \gamma_K = \alpha γ K = α K ( 0 , v ) = k 0 K(0,v) = k_0 K ( 0 , v ) = k 0 v v v γ K = β \gamma_K = \beta γ K = β K ( u , 0 ) = k 0 K(u,0) = k_0 K ( u , 0 ) = k 0 u u u γ K = min ( α , β ) \gamma_K = \min(\alpha,\beta) γ K = min ( α , β ) C K C_K C K n n n ϵ \epsilon ϵ n n n P P P
∥ K P − k 0 ∥ ∞ ≤ C K ( ∥ u ∥ ∞ + ∥ v ∥ ∞ ) = o ( δ n ) , ∥ K P − k 0 ∥ C t ≤ C K ϵ ( 0 < t ≤ γ K ) , \lVert K_P-k_0\rVert_\infty\leq C_K(\lVert u\rVert_\infty+\lVert v\rVert_\infty)=o(\delta_n), \qquad\lVert K_P-k_0\rVert_{C^t}\leq C_K\epsilon\quad(0<t\leq\gamma_K), ∥ K P − k 0 ∥ ∞ ≤ C K (∥ u ∥ ∞ + ∥ v ∥ ∞ ) = o ( δ n ) , ∥ K P − k 0 ∥ C t ≤ C K ϵ ( 0 < t ≤ γ K ) , and, if k 0 > 0 k_0>0 k 0 > 0 r − k 0 ≤ p K P ≤ r + k 0 r-k_0\leq pK_P\leq r+k_0 r − k 0 ≤ p K P ≤ r + k 0 P t P_t P t K P t K_{P_t} K P t K ( u ∗ , t ) K(u_*,t) K ( u ∗ , t ) k 0 k_0 k 0 ( u ∗ , t ) → 0 (u_*,t)\to0 ( u ∗ , t ) → 0
Consider finitely many conditions on ( p , u , v ) (p,u,v) ( p , u , v ) C ∞ C^\infty C ∞ K K K k 0 = K ( 0 , 0 ) k_0=K(0,0) k 0 = K ( 0 , 0 )
(H) ∥ K P ∥ C t ≤ H ′ \lVert K_P\rVert_{C^t}\leq H' ∥ K P ∥ C t ≤ H ′ 0 < t ≤ γ K 0<t\leq\gamma_K 0 < t ≤ γ K ∣ k 0 ∣ < H ′ \lvert k_0\rvert<H' ∣ k 0 ∣ < H ′
(I) k − ≤ K P ≤ k + k_-\leq K_P\leq k_+ k − ≤ K P ≤ k + k − < k 0 < k + k_-<k_0<k_+ k − < k 0 < k +
(W) g − ′ ≤ p K P ≤ g + ′ g'_-\leq pK_P\leq g'_+ g − ′ ≤ p K P ≤ g + ′ k 0 > 0 k_0>0 k 0 > 0 g − ′ ≤ r − k 0 g'_-\leq r-k_0 g − ′ ≤ r − k 0 r + k 0 ≤ g + ′ r+k_0\leq g'_+ r + k 0 ≤ g + ′
If ϵ \epsilon ϵ n n n u ∗ u_* u ∗ P t P_t P t p = 1 p=1 p = 1 t t t θ < 1 / 2 \theta<1/2 θ < 1/2
Proof. Shrinking the domain of K K K N N N ( u , 0 ) (u,0) ( u , 0 ) ( 0 , v ) (0,v) ( 0 , v ) N N N ( u , v ) (u,v) ( u , v ) K K K N N N C K C_K C K ∥ u ∥ ∞ + ∥ v ∥ ∞ ≤ C n − min ( α , β ) / d = o ( δ n ) \lVert u\rVert_\infty+\lVert v\rVert_\infty\leq Cn^{-\min(\alpha,\beta)/d}=o(\delta_n) ∥ u ∥ ∞ + ∥ v ∥ ∞ ≤ C n − m i n ( α , β ) / d = o ( δ n )
For the Hölder bounds, we write
K ( u , v ) − k 0 = { K ( u , v ) − K ( 0 , v ) } + { K ( 0 , v ) − k 0 } K(u,v)-k_0=\{K(u,v)-K(0,v)\}+\{K(0,v)-k_0\} K ( u , v ) − k 0 = { K ( u , v ) − K ( 0 , v )} + { K ( 0 , v ) − k 0 } on N N N u = 0 u=0 u = 0 v = 0 v=0 v = 0 (88) , for all large n n n ( u , v ) (u,v) ( u , v ) C ϵ C\epsilon C ϵ C α C^\alpha C α C β C^\beta C β C C C ϵ \epsilon ϵ K ( 0 , v ) = k 0 K(0,v)=k_0 K ( 0 , v ) = k 0 C α C^\alpha C α K ( u , 0 ) = k 0 K(u,0)=k_0 K ( u , 0 ) = k 0 K − k 0 K-k_0 K − k 0 v = 0 v=0 v = 0 C β C^\beta C β (88) . In general, both braces are bounded in C min ( α , β ) C^{\min(\alpha,\beta)} C m i n ( α , β ) ∥ f ∥ C t ≤ ( 1 + d ) ∥ f ∥ C t ′ \lVert f\rVert_{C^t}\leq(1+d)\lVert f\rVert_{C^{t'}} ∥ f ∥ C t ≤ ( 1 + d ) ∥ f ∥ C t ′ 0 < t ≤ t ′ 0<t\leq t' 0 < t ≤ t ′ f f f [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d ∣ x − y ∣ ≤ d \lvert x-y\rvert\leq\sqrt{d} ∣ x − y ∣ ≤ d [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d ∣ t ∣ < ∣ t ′ ∣ \lvert t\rvert<\lvert t'\rvert ∣ t ∣ < ∣ t ′ ∣ ⌈ t ⌉ − 1 \lceil t\rceil-1 ⌈ t ⌉ − 1 d ∥ f ∥ C t ′ \sqrt{d}\lVert f\rVert_{C^{t'}} d ∥ f ∥ C t ′ [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d C K C_K C K 0 < t ≤ γ K 0<t\leq\gamma_K 0 < t ≤ γ K
Let k 0 > 0 k_0>0 k 0 > 0 η = ∥ K P − k 0 ∥ ∞ \eta=\lVert K_P-k_0\rVert_\infty η = ∥ K P − k 0 ∥ ∞ η = o ( δ n ) \eta=o(\delta_n) η = o ( δ n ) n n n K P > 0 K_P>0 K P > 0 r + η ≤ k 0 δ n r_+\eta\le k_0\delta_n r + η ≤ k 0 δ n (71) gives
p K P ≥ ( r − + δ n ) ( k 0 − η ) ≥ r − k 0 + k 0 δ n − r + η ≥ r − k 0 , pK_P\ge(r_-+\delta_n)(k_0-\eta)\ge r_-k_0+k_0\delta_n-r_+\eta\ge r_-k_0, p K P ≥ ( r − + δ n ) ( k 0 − η ) ≥ r − k 0 + k 0 δ n − r + η ≥ r − k 0 , and in the same way p K P ≤ ( r + − δ n ) ( k 0 + η ) ≤ r + k 0 − k 0 δ n + r + η ≤ r + k 0 pK_P\le(r_+-\delta_n)(k_0+\eta)\le r_+k_0-k_0\delta_n+r_+\eta\le r_+k_0 p K P ≤ ( r + − δ n ) ( k 0 + η ) ≤ r + k 0 − k 0 δ n + r + η ≤ r + k 0
Since a constant has Hölder norm equal to its absolute value, we obtain (H) from ∥ K P ∥ C t ≤ ∣ k 0 ∣ + C K ϵ ≤ H ′ \lVert K_P\rVert_{C^t}\le|k_0|+C_K\epsilon\le H' ∥ K P ∥ C t ≤ ∣ k 0 ∣ + C K ϵ ≤ H ′ C K ϵ ≤ H ′ − ∣ k 0 ∣ C_K\epsilon\le H'-|k_0| C K ϵ ≤ H ′ − ∣ k 0 ∣ p K P pK_P p K P P t P_t P t p = 1 p=1 p = 1 K P t = K ( u ∗ , t ) K_{P_t}=K(u_*,t) K P t = K ( u ∗ , t ) ( u ∗ , t ) = ( 0 , 0 ) (u_*,t)=(0,0) ( u ∗ , t ) = ( 0 , 0 ) g − ≤ r − k 0 < k 0 < r + k 0 ≤ g + g_-\le r_-k_0<k_0<r_+k_0\le g_+ g − ≤ r − k 0 < k 0 < r + k 0 ≤ g + ( u ∗ , t ) (u_*,t) ( u ∗ , t )
Proof of Theorem 2.3(b). Since the minimax risk and the minimax error probability do not decrease when the class grows, it suffices to work on P r ( π 0 ) \mathcal{P}_r(\pi_0) P r ( π 0 ) Σ − 1 r \Sigma^{-1}r Σ − 1 r (8) , and the diagonal score r d / E π 0 r d 2 r_d/\mathbb{E}_{\pi_0}r_d^2 r d / E π 0 r d 2 (8) . Let r − = g − / w 0 r_-=g_-/w_0 r − = g − / w 0 r + = g + / w 0 r_+=g_+/w_0 r + = g + / w 0 0 < r − < 1 < r + 0<r_-<1<r_+ 0 < r − < 1 < r + (8) . Since scaling both endpoints of an interval by the same factor leaves ρ \rho ρ a ∗ + b ∗ = θ a_*+b_*=\theta a ∗ + b ∗ = θ τ \tau τ a η a_\eta a η Proposition 7.1 are those of (10) and (12) .
Under a hard law or a law P t P_t P t w , a , b w,a,b w , a , b K P K_P K P w ( u , v ) , a ( u , v ) , b ( u , v ) w(u,v),a(u,v),b(u,v) w ( u , v ) , a ( u , v ) , b ( u , v ) (85) and (86) , with k 0 = w 0 , a 0 , b 0 k_0=w_0,a_0,b_0 k 0 = w 0 , a 0 , b 0 g = p w g=pw g = pw P r ( π 0 ) \mathcal{P}_r(\pi_0) P r ( π 0 ) max ( δ , w 0 − r ) ≤ w ≤ w 0 + r \max(\delta,w_0-r)\le w\le w_0+r max ( δ , w 0 − r ) ≤ w ≤ w 0 + r ∣ a − a 0 ∣ ≤ r |a-a_0|\le r ∣ a − a 0 ∣ ≤ r ∣ b − b 0 ∣ ≤ r |b-b_0|\le r ∣ b − b 0 ∣ ≤ r g − ≤ p w ≤ g + g_-\le pw\le g_+ g − ≤ pw ≤ g + r ± w 0 = g ± r_\pm w_0=g_\pm r ± w 0 = g ± ∥ a ∥ C α ≤ H \lVert a\rVert_{C^\alpha}\le H ∥ a ∥ C α ≤ H ∥ b ∥ C β ≤ H \lVert b\rVert_{C^\beta}\le H ∥ b ∥ C β ≤ H (8) , the baseline values satisfy the conditions of types (I) and (H) strictly. For (H), we may take γ K = α \gamma_K=\alpha γ K = α a a a γ K = β \gamma_K=\beta γ K = β b b b a ( 0 , v ) = a 0 a(0,v)=a_0 a ( 0 , v ) = a 0 b ( u , 0 ) = b 0 b(u,0)=b_0 b ( u , 0 ) = b 0 α = β \alpha=\beta α = β ϵ \epsilon ϵ Lemma 7.2 for these conditions. Then, for all large n n n P r ( π 0 ) \mathcal{P}_r(\pi_0) P r ( π 0 ) u ∗ ≠ 0 u_*\ne0 u ∗ = 0 P t P_t P t t t t
If 0 < θ < 1 / 2 0<\theta<1/2 0 < θ < 1/2 Proposition 7.1 (a) with P = P r ( π 0 ) \mathcal{P}=\mathcal{P}_r(\pi_0) P = P r ( π 0 ) T = ψ T=\psi T = ψ t ∗ = 0 t_*=0 t ∗ = 0 c 0 > 0 c_0>0 c 0 > 0 n n n c 0 a n log n c_0a_n\log n c 0 a n log n 3 / 8 3/8 3/8 r n ( ψ , P r ( π 0 ) ) ≥ 3 / 8 c 0 a n log n r_n(\psi,\mathcal{P}_r(\pi_0))\ge\sqrt{3/8}\,c_0a_n\log n r n ( ψ , P r ( π 0 )) ≥ 3/8 c 0 a n log n c = c 0 / 2 c=c_0/2 c = c 0 /2 (14) and the lower bound in (16) , since 3 / 8 ≥ 1 / 4 3/8\ge1/4 3/8 ≥ 1/4 3 / 8 ≥ 1 / 2 \sqrt{3/8}\ge1/2 3/8 ≥ 1/2 θ > 0 \theta>0 θ > 0 Proposition 7.1 (b), with P = P r ( π 0 ) \mathcal{P}=\mathcal{P}_r(\pi_0) P = P r ( π 0 ) T = ψ T=\psi T = ψ c ′ > 0 c'>0 c ′ > 0 r n ( ψ , P r ( π 0 ) ) ≥ c ′ n − 1 / 2 r_n(\psi,\mathcal{P}_r(\pi_0))\ge c'n^{-1/2} r n ( ψ , P r ( π 0 )) ≥ c ′ n − 1/2 n n n (15) when θ ≥ 1 / 2 \theta\ge1/2 θ ≥ 1/2 θ ≥ 1 / 2 \theta\ge1/2 θ ≥ 1/2 c = c ′ c=c' c = c ′ n n n 0 < θ < 1 / 2 0<\theta<1/2 0 < θ < 1/2 c = min { c 0 / 2 , c ′ } c=\min\{c_0/2,c'\} c = min { c 0 /2 , c ′ } n n n
Conclusion
First, when s < d / 4 s < d/4 s < d /4 Theorem 2.3 differ by the factor ( log n ) ν / 2 − 3 / 4 (\log n)^{\nu/2-3/4} ( log n ) ν /2 − 3/4 ( log n ) 1 / 4 (\log n)^{1/4} ( log n ) 1/4 0 < α , β ≤ 1 0 < \alpha,\beta\le1 0 < α , β ≤ 1 ( log n ) 1 / 4 (\log n)^{1/4} ( log n ) 1/4 § 2.3 ). Each of our lower-bound constructions, in contrast, places its bumps at a single resolution. Closing this gap would determine the minimax rate up to constants. The power of log n \log n log n [46 ] .
Second, our estimator uses α , β \alpha,\beta α , β g − , g + g_{-},g_{+} g − , g + [43 ] and Park [46, Corollary 1] attain the rough-design exponent for the expected conditional covariance without knowing the nuisance smoothness, and adaptive results for related functionals under smooth density assumptions were obtained in [39 ] . Park also reports an estimator that does not use α , β \alpha,\beta α , β log n \log n log n { α + β < d / 2 } \{\alpha+ \beta< d/2\} { α + β < d /2 } [46 Section 3.4] . This estimator is analyzed in Theorem S13 of the supplement of [46 ] . For the general class, it is open whether the upper bound of Theorem 2.3 , including the correction, can be attained without knowledge of α , β \alpha,\beta α , β [46 ] , it is open whether this is possible without the larger power of log n \log n log n
Third, minimax rates remain unresolved when g g g [76 Remark 5] . When the two nuisance functions have equal smoothness indices, the conjecture following equation (1.3) of Robins et al. [48 ] addresses this case; our results confirm the exponent it suggests in the rough-design limit.
Reductions
Most targets of Section 3 are related to a target of Theorem 2.3 through an observable mean, a Lipschitz map, or a transformation of the data that does not depend on the unknown law. We record the lower-bound constructions and these three reductions.
Hard laws
Definition A.1 (Hardness). Let P n \mathcal{P}_n P n T T T P n \mathcal{P}_n P n t n > 0 t_n > 0 t n > 0 T T T hard at scale t n t_n t n on ( P n ) (\mathcal{P}_n) ( P n )
lim inf n → ∞ inf T ^ sup P ∈ P n P P { ∣ T ^ − T ( P ) ∣ ≥ t n } ≥ 3 8 . \liminf_{n \to\infty} \inf_{\widehat{T}} \sup_{P \in\mathcal{P}_n} \mathbb{P}_P\{|\widehat{T} - T(P)| \ge t_n\} \ge\frac{3}{8}. n → ∞ lim inf T inf P ∈ P n sup P P { ∣ T − T ( P ) ∣ ≥ t n } ≥ 8 3 . where, for each n n n n n n
Hardness at scale t n t_n t n c t n ct_n c t n c ∈ ( 0 , 1 ] c \in(0,1] c ∈ ( 0 , 1 ] P n ′ ⊇ P n \mathcal{P}'_n \supseteq\mathcal{P}_n P n ′ ⊇ P n
Lemma A.2 (From hardness to risk). Let T T T hard at scale t n t_n t n on ( P n ) (\mathcal{P}_n) ( P n ) P \mathcal{P} P P n ⊆ P \mathcal{P}_n \subseteq\mathcal{P} P n ⊆ P n n n n n n
inf T ^ sup P ∈ P P P { ∣ T ^ − T ( P ) ∣ ≥ t n } ≥ 1 4 , r n ( T , P ) ≥ t n 2 . (89) \inf_{\widehat{T}} \sup_{P \in\mathcal{P}} \mathbb{P}_P\{|\widehat{T} - T(P)| \ge t_n\} \ge\frac{1}{4}, \qquad r_n(T,\mathcal{P}) \ge\frac{t_n}{2}. \tag*{(89)} T inf P ∈ P sup P P { ∣ T − T ( P ) ∣ ≥ t n } ≥ 4 1 , r n ( T , P ) ≥ 2 t n . (89) The infimum includes randomized estimators. In particular, if 0 < θ < 1 / 2 0 < \theta< 1/2 0 < θ < 1/2 t n = c 1 a n ( θ , τ I ) t_n = c_1a_n(\theta,\tau_I) t n = c 1 a n ( θ , τ I ) c 1 > 0 c_1 > 0 c 1 > 0 I I I Definition 3.1 , then ( T , P ) (T,\mathcal{P}) ( T , P ) ( θ , I ) (\theta,I) ( θ , I ) c = c 1 / 2 c = c_1/2 c = c 1 /2
Proof. By the lim inf \liminf lim inf P n \mathcal{P}_n P n P ⊇ P n \mathcal{P} \supseteq\mathcal{P}_n P ⊇ P n sup P ∈ P E P ( T ^ − T ( P ) ) 2 ≥ t n 2 sup P ∈ P P P { ∣ T ^ − T ( P ) ∣ ≥ t n } \sup_{P \in\mathcal{P}} \mathbb{E}_P(\widehat{T} - T(P))^2 \ge t_n^2 \sup_{P \in\mathcal{P}} \mathbb{P}_P\{|\widehat{T} - T(P)| \ge t_n\} sup P ∈ P E P ( T − T ( P ) ) 2 ≥ t n 2 sup P ∈ P P P { ∣ T − T ( P ) ∣ ≥ t n } t n = c 1 a n = 2 c a n t_n = c_1a_n = 2c\,a_n t n = c 1 a n = 2 c a n
We fix the inputs of Proposition 7.1 : a finite-support law π 0 \pi_0 π 0 D , U , V , W D,U,V,W D , U , V , W λ \lambda λ π 0 \pi_0 π 0 s d s_d s d E π 0 s d = 0 \mathbb{E}_{\pi_0}s_{\mathrm d}=0 E π 0 s d = 0 E π 0 [ ( U − a 0 D ) s d ] = 1 \mathbb{E}_{\pi_0}[(U-a_0D)s_{\mathrm d}]=1 E π 0 [( U − a 0 D ) s d ] = 1 α , β \alpha,\beta α , β θ = ( α + β ) / d < 1 / 2 \theta=(\alpha+\beta)/d<1/2 θ = ( α + β ) / d < 1/2 [ r − , r + ] [r_-,r_+] [ r − , r + ] 0 < r − < 1 < r + 0<r_-<1<r_+ 0 < r − < 1 < r + ϵ > 0 \epsilon>0 ϵ > 0 Proposition 7.1 (a), there exists a fixed c 1 > 0 c_1>0 c 1 > 0 t ∗ t_* t ∗ ψ + t ∗ \psi+t_* ψ + t ∗ c 1 a n ( θ , τ ) log n c_1a_n(\theta,\tau)\log n c 1 a n ( θ , τ ) log n ( P n h ) (\mathcal{P}_n^{\mathrm h}) ( P n h ) τ \tau τ [ r − , r + ] [r_-,r_+] [ r − , r + ] ψ \psi ψ § 3 for ( D , U , V , W , λ ) (D,U,V,W,\lambda) ( D , U , V , W , λ ) x x x π 0 \pi_0 π 0 ( u ( x ) , v ( x ) ) (u(x),v(x)) ( u ( x ) , v ( x )) ∥ u ∥ ∞ ≤ C n − α / d \|u\|_\infty\leq Cn^{-\alpha/d} ∥ u ∥ ∞ ≤ C n − α / d ∥ v ∥ ∞ ≤ C n − β / d \|v\|_\infty\leq Cn^{-\beta/d} ∥ v ∥ ∞ ≤ C n − β / d Lemma 7.2 , we take ϵ \epsilon ϵ n n n P t P_t P t Proposition 7.1 (b) satisfy them for every sufficiently small u ∗ u_* u ∗ t t t Lemma A.2 .
Three reductions
Let h h h R k \mathbb{R}^k R k k ≥ 0 k\geq0 k ≥ 0 μ ( P ) = E P h ( O ) \mu(P)=\mathbb{E}_P h(O) μ ( P ) = E P h ( O ) R \mathcal{R} R μ ( P ) \mu(P) μ ( P ) P P P b h 2 = sup P E P ∥ h ( O ) − μ ( P ) ∥ 2 2 b_h^2=\sup_P\mathbb{E}_P\|h(O)-\mu(P)\|_2^2 b h 2 = sup P E P ∥ h ( O ) − μ ( P ) ∥ 2 2 k = 0 k=0 k = 0 h h h μ \mu μ b h = 0 b_h=0 b h = 0 I 1 , … , I m I_1,\ldots,I_m I 1 , … , I m Φ \Phi Φ I 1 × ⋯ × I m × R I_1\times\cdots\times I_m\times\mathcal{R} I 1 × ⋯ × I m × R L L L L ≥ 1 L\geq1 L ≥ 1 R \mathcal{R} R
Lemma A.3 (Reductions).
(a) (Combination. ) Suppose that T 1 , … , T m T_1,\ldots,T_m T 1 , … , T m P \mathcal{P} P T i ( P ) ∈ I i T_i(P)\in I_i T i ( P ) ∈ I i T 0 ( P ) = Φ ( T 1 ( P ) , … , T m ( P ) , μ ( P ) ) T_0(P)=\Phi(T_1(P),\ldots,T_m(P),\mu(P)) T 0 ( P ) = Φ ( T 1 ( P ) , … , T m ( P ) , μ ( P )) P ∈ P P\in\mathcal{P} P ∈ P Φ \Phi Φ L L L n ≥ 1 n\geq1 n ≥ 1
r n ( T 0 , P ) ≤ L ∑ i = 1 m r n ( T i , P ) + L b h n − 1 / 2 . r_n(T_0,\mathcal{P})\leq L\sum_{i=1}^{m}r_n(T_i,\mathcal{P})+Lb_hn^{-1/2}. r n ( T 0 , P ) ≤ L i = 1 ∑ m r n ( T i , P ) + L b h n − 1/2 . (b) (Reverse relations. ) Let m = 1 m=1 m = 1 P ′ \mathcal{P}' P ′ T 0 ( P ) ∈ I 1 T_0(P)\in I_1 T 0 ( P ) ∈ I 1
∣ T 1 ( P ) − Φ ( T 0 ( P ) , μ ( P ) ) ∣ ≤ e ( P ∈ P ′ ) , (90) \lvert T_1(P)-\Phi(T_0(P),\mu(P))\rvert\leq e\qquad(P\in\mathcal{P}'), \tag*{(90)} ∣ T 1 ( P ) − Φ ( T 0 ( P ) , μ ( P ))∣ ≤ e ( P ∈ P ′ ) , (90) where Φ \Phi Φ L L L e ≥ 0 e\geq0 e ≥ 0 n ≥ 1 n\geq1 n ≥ 1 t > 0 t>0 t > 0 t ≥ 2 e t\geq2e t ≥ 2 e
r n ( T 1 , P ′ ) ≤ L r n ( T 0 , P ′ ) + L b h n − 1 / 2 + e , (91) r_n(T_1,\mathcal{P}')\leq Lr_n(T_0,\mathcal{P}')+Lb_hn^{-1/2}+e, \tag*{(91)} r n ( T 1 , P ′ ) ≤ L r n ( T 0 , P ′ ) + L b h n − 1/2 + e , (91) inf T ^ 0 sup P ∈ P ′ P P { ∣ T ^ 0 − T 0 ( P ) ∣ ≥ t 4 L } ≥ inf T ^ 1 sup P ∈ P ′ P P { ∣ T ^ 1 − T 1 ( P ) ∣ ≥ t } − 16 L 2 b h 2 n t 2 . (92) \inf_{\widehat{T}_0}\sup_{P\in\mathcal{P}'}\mathbb{P}_P\left\{\lvert\widehat{T}_0-T_0(P)\rvert\geq\frac{t}{4L}\right\}
\geq
\inf_{\widehat{T}_1}\sup_{P\in\mathcal{P}'}\mathbb{P}_P\{\lvert\widehat{T}_1-T_1(P)\rvert\geq t\}
-\frac{16L^2b_h^2}{nt^2}. \tag*{(92)} T 0 inf P ∈ P ′ sup P P { ∣ T 0 − T 0 ( P )∣ ≥ 4 L t } ≥ T 1 inf P ∈ P ′ sup P P {∣ T 1 − T 1 ( P )∣ ≥ t } − n t 2 16 L 2 b h 2 . (92) Consequently, suppose that for all large n n n P ′ = P n ′ \mathcal{P}'=\mathcal{P}'_n P ′ = P n ′ e = e n ≤ t n / 2 e=e_n\leq t_n/2 e = e n ≤ t n /2 h h h R \mathcal{R} R I 1 I_1 I 1 Φ \Phi Φ n n n T 1 T_1 T 1 t n t_n t n ( P n ′ ) (\mathcal{P}'_n) ( P n ′ ) k = 0 k=0 k = 0 n t n 2 → ∞ nt_n^2\to\infty n t n 2 → ∞ T 0 T_0 T 0 t n / ( 4 L ) t_n/(4L) t n / ( 4 L ) ( P n ′ ) (\mathcal{P}'_n) ( P n ′ )
(c) (Kernels. ) Let K K K P 0 \mathcal{P}_0 P 0 P 1 \mathcal{P}_1 P 1 K P KP K P P P P K P ∈ P 1 KP \in\mathcal{P}_1 K P ∈ P 1 T 1 ( K P ) = T 0 ( P ) T_1(KP)=T_0(P) T 1 ( K P ) = T 0 ( P ) P ∈ P 0 P \in\mathcal{P}_0 P ∈ P 0 n ≥ 1 n \ge1 n ≥ 1 t > 0 t > 0 t > 0
inf T ^ 0 sup P ∈ P 0 P P { ∣ T ^ 0 − T 0 ( P ) ∣ ≥ t } ≤ inf T ^ 1 sup Q ∈ P 1 P Q { ∣ T ^ 1 − T 1 ( Q ) ∣ ≥ t } . \inf_{\widehat{T}_0}\sup_{P\in\mathcal{P}_0}\mathbb{P}_P\{|\widehat{T}_0-T_0(P)|\ge t\}\le\inf_{\widehat{T}_1}\sup_{Q\in\mathcal{P}_1}\mathbb{P}_Q\{|\widehat{T}_1-T_1(Q)|\ge t\}. T 0 inf P ∈ P 0 sup P P { ∣ T 0 − T 0 ( P ) ∣ ≥ t } ≤ T 1 inf Q ∈ P 1 sup P Q { ∣ T 1 − T 1 ( Q ) ∣ ≥ t } . and r n ( T 0 , P 0 ) ≤ r n ( T 1 , P 1 ) r_n(T_0,\mathcal{P}_0)\le r_n(T_1,\mathcal{P}_1) r n ( T 0 , P 0 ) ≤ r n ( T 1 , P 1 ) T 1 T_1 T 1 P 1 \mathcal{P}_1 P 1 T 0 T_0 T 0 P 0 \mathcal{P}_0 P 0 T 0 T_0 T 0 P 0 \mathcal{P}_0 P 0 T 1 T_1 T 1 P 1 \mathcal{P}_1 P 1 T 0 T_0 T 0 t n t_n t n ( P 0 , n ) (\mathcal{P}_0,n) ( P 0 , n ) T 1 T_1 T 1 t n t_n t n ( K P 0 , n ) (K\mathcal{P}_0,n) ( K P 0 , n )
Proof. Let μ ^ \widehat{\mu} μ R \mathcal{R} R h ( O 1 ) , … , h ( O n ) h(O_1),\ldots,h(O_n) h ( O 1 ) , … , h ( O n ) Π i \Pi_i Π i I i I_i I i R \mathcal{R} R I i I_i I i E P ∥ μ ^ − μ ( P ) ∥ 2 2 ≤ b h 2 / n \mathbb{E}_P\|\widehat{\mu}-\mu(P)\|_2^2\le b_h^2/n E P ∥ μ − μ ( P ) ∥ 2 2 ≤ b h 2 / n S , S 1 , … , S m S,S_1,\ldots,S_m S , S 1 , … , S m S i ( P ) ∈ I i S_i(P)\in I_i S i ( P ) ∈ I i ∣ S ( P ) − Φ ( S 1 ( P ) , … , S m ( P ) , μ ( P ) ) ∣ ≤ e |S(P)-\Phi(S_1(P),\ldots,S_m(P),\mu(P))|\le e ∣ S ( P ) − Φ ( S 1 ( P ) , … , S m ( P ) , μ ( P )) ∣ ≤ e S ^ i \widehat{S}_i S i S ^ = Φ ( Π 1 S ^ 1 , … , Π m S ^ m , μ ^ ) \widehat{S}=\Phi(\Pi_1\widehat{S}_1,\ldots,\Pi_m\widehat{S}_m,\widehat{\mu}) S = Φ ( Π 1 S 1 , … , Π m S m , μ )
∣ S ^ − S ( P ) ∣ ≤ L { ∑ i = 1 m ∣ S ^ i − S i ( P ) ∣ + ∥ μ ^ − μ ( P ) ∥ 2 } + e . |\widehat{S}-S(P)|\le L\left\{\sum_{i=1}^{m}|\widehat{S}_i-S_i(P)|+\|\widehat{\mu}-\mu(P)\|_2\right\}+e. ∣ S − S ( P ) ∣ ≤ L { i = 1 ∑ m ∣ S i − S i ( P ) ∣ + ∥ μ − μ ( P ) ∥ 2 } + e . With S = T 0 S=T_0 S = T 0 S i = T i S_i=T_i S i = T i e = 0 e=0 e = 0 L 2 ( P n ) L^2(P^n) L 2 ( P n ) m = 1 m=1 m = 1 S = T 1 S=T_1 S = T 1 S 1 = T 0 S_1=T_0 S 1 = T 0 P ′ \mathcal{P}' P ′ ∣ S ^ − T 1 ( P ) ∣ ≥ t ≥ 2 e |\widehat{S}-T_1(P)|\ge t\ge2e ∣ S − T 1 ( P ) ∣ ≥ t ≥ 2 e ∣ T ^ 0 − T 0 ( P ) ∣ ≥ t / ( 4 L ) |\widehat{T}_0-T_0(P)|\ge t/(4L) ∣ T 0 − T 0 ( P ) ∣ ≥ t / ( 4 L ) ∥ μ ^ − μ ( P ) ∥ 2 ≥ t / ( 4 L ) \|\widehat{\mu}-\mu(P)\|_2\ge t/(4L) ∥ μ − μ ( P ) ∥ 2 ≥ t / ( 4 L ) 16 L 2 b h 2 / ( n t 2 ) 16L^2b_h^2/(nt^2) 16 L 2 b h 2 / ( n t 2 ) S ^ \widehat{S} S T 1 T_1 T 1 P ∈ P ′ P\in\mathcal{P}' P ∈ P ′ T ^ 0 \widehat{T}_0 T 0 t = t n t=t_n t = t n lim inf \liminf lim inf k = 0 k=0 k = 0
For (c), we apply K K K n n n T ^ 1 \widehat{T}_1 T 1 P P P ( K P ) n (KP)^n ( K P ) n T 1 ( K P ) = T 0 ( P ) T_1(KP)=T_0(P) T 1 ( K P ) = T 0 ( P ) T ^ 1 \widehat{T}_1 T 1 K P ∈ P 1 KP\in\mathcal{P}_1 K P ∈ P 1
Lemma A.3(c) is the elementary direction of Le Cam’s comparison of experiments by randomization [32 Chapter 2] : an experiment is at least as informative as its image under a Markov kernel that does not depend on the unknown law. Composing an estimator for the image with the kernel, we obtain an estimator whose error has the same law. This gives both the inequality for error probabilities and the inequality for risks.
For a ratio we use (a) with Φ ( x , y ) = x / y \Phi(x,y)=x/y Φ ( x , y ) = x / y [ − C 0 , C 0 ] × [ δ D , C 0 ] [-C_0,C_0]\times[\delta_D,C_0] [ − C 0 , C 0 ] × [ δ D , C 0 ] L L L L = max { 1 , δ D − 1 , C 0 δ D − 2 } L=\max\{1,\delta_D^{-1},C_0\delta_D^{-2}\} L = max { 1 , δ D − 1 , C 0 δ D − 2 } ∣ x / y − x ′ / y ′ ∣ ≤ ∣ x − x ′ ∣ / δ D + C 0 ∣ y − y ′ ∣ / δ D 2 |x/y-x'/y'|\le|x-x'|/\delta_D+C_0|y-y'|/\delta_D^2 ∣ x / y − x ′ / y ′ ∣ ≤ ∣ x − x ′ ∣/ δ D + C 0 ∣ y − y ′ ∣/ δ D 2
Missing-data means and treatment effects
The treatment effects inherit lower bounds below the threshold from one MAR family through a kernel with the observed regression in one arm and the constant 1 / 2 1/2 1/2
The MAR family and two maps
We take Z = ( R , R Y ) Z=(R,RY) Z = ( R , R Y ) R , Y ∈ { 0 , 1 } R,Y\in\{0,1\} R , Y ∈ { 0 , 1 } ( D , U , V , W , λ ) = ( R , 1 , R Y , 0 , 1 ) (D,U,V,W,\lambda)=(R,1,RY,0,1) ( D , U , V , W , λ ) = ( R , 1 , R Y , 0 , 1 ) π 0 \pi_0 π 0 ( R , R Y ) = ( 0 , 0 ) , ( 1 , 0 ) , ( 1 , 1 ) (R,RY)=(0,0),(1,0),(1,1) ( R , R Y ) = ( 0 , 0 ) , ( 1 , 0 ) , ( 1 , 1 ) 1 / 2 , 1 / 4 , 1 / 4 1/2,1/4,1/4 1/2 , 1/4 , 1/4 ( w 0 , a 0 , b 0 ) = ( 1 / 2 , 2 , 1 / 2 ) (w_0,a_0,b_0)=(1/2,2,1/2) ( w 0 , a 0 , b 0 ) = ( 1/2 , 2 , 1/2 ) r = ( 1 − 2 R , R Y − R / 2 ) T \mathbf{r}=(1-2R,RY-R/2)^{\mathsf T} r = ( 1 − 2 R , R Y − R /2 ) T ( 1 , 0 ) , ( − 1 , − 1 / 2 ) (1,0),(-1,-1/2) ( 1 , 0 ) , ( − 1 , − 1/2 ) ( − 1 , 1 / 2 ) (-1,1/2) ( − 1 , 1/2 ) Σ = Cov π 0 ( r ) = diag ( 1 , 1 / 8 ) \Sigma=\operatorname{Cov}_{\pi_0}(\mathbf{r})=\operatorname{diag}(1,1/8) Σ = Cov π 0 ( r ) = diag ( 1 , 1/8 ) Σ \Sigma Σ ( s u , s v ) T = Σ − 1 r = ( 1 − 2 R , 8 R Y − 4 R ) T (s_u,s_v)^{\mathsf T}=\Sigma^{-1}\mathbf{r}=(1-2R,8RY-4R)^{\mathsf T} ( s u , s v ) T = Σ − 1 r = ( 1 − 2 R , 8 R Y − 4 R ) T π 0 \pi_0 π 0 § 8.1 . The perturbed response law π u , v ( { z } ) = π 0 ( { z } ) { 1 + u s u ( z ) + v s v ( z ) } \pi_{u,v}(\{z\})=\pi_0(\{z\})\{1+us_u(z)+vs_v(z)\} π u , v ({ z }) = π 0 ({ z }) { 1 + u s u ( z ) + v s v ( z )}
P ( R = 0 ∣ X ) = 1 + u 2 , P ( R = 1 , Y = 0 ∣ X ) = 1 − u 4 − v , P ( R = 1 , Y = 1 ∣ X ) = 1 − u 4 + v . (93) \begin{aligned}
P(R=0\mid X)&=\frac{1+u}{2},\\
P(R=1,Y=0\mid X)&=\frac{1-u}{4}-v,\qquad P(R=1,Y=1\mid X)=\frac{1-u}{4}+v. \tag*{(93)}
\end{aligned} P ( R = 0 ∣ X ) P ( R = 1 , Y = 0 ∣ X ) = 2 1 + u , = 4 1 − u − v , P ( R = 1 , Y = 1 ∣ X ) = 4 1 − u + v . (93) so that
w = 1 − u 2 , 1 w = 2 1 − u , 1 1 − w = 2 1 + u , b = 1 2 + 2 v 1 − u . (94) w=\frac{1-u}{2},\qquad\frac{1}{w}=\frac{2}{1-u},\qquad\frac{1}{1-w}=\frac{2}{1+u},\qquad b=\frac{1}{2}+\frac{2v}{1-u}. \tag*{(94)} w = 2 1 − u , w 1 = 1 − u 2 , 1 − w 1 = 1 + u 2 , b = 2 1 + 1 − u 2 v . (94) and ψ ( P ) = ∫ a b g = E P [ b ( X ) ] \psi(P)=\int abg=\mathbb{E}_P[b(X)] ψ ( P ) = ∫ ab g = E P [ b ( X )] Y Y Y b ( X ) b(X) b ( X ) R R R X X X [49 Section 3] uses binary models of this kind.
For indices ( α , β ′ ) (\alpha,\beta') ( α , β ′ ) ( α + β ′ ) / d < 1 / 2 (\alpha+\beta')/d<1/2 ( α + β ′ ) / d < 1/2 [ r − , r + ] [r_-,r_+] [ r − , r + ] 0 < r − < 1 < r + 0<r_-<1<r_+ 0 < r − < 1 < r + § 10 .1 for this π 0 \pi_0 π 0 c 1 a n ( ( α + β ′ ) / d , τ ) c_1a_n((\alpha+\beta')/d,\tau) c 1 a n (( α + β ′ ) / d , τ ) τ \tau τ [ r − , r + ] [r_-,r_+] [ r − , r + ] Lemma 7.2 , the functions w w w 1 − w 1-w 1 − w 1 / w 1/w 1/ w 1 / ( 1 − w ) 1/(1-w) 1/ ( 1 − w ) K P K_P K P K ( 0 , v ) K(0,v) K ( 0 , v ) γ K = α \gamma_K=\alpha γ K = α b b b K P K_P K P K ( u , 0 ) = 1 / 2 K(u,0)=1/2 K ( u , 0 ) = 1/2 γ K = β ′ \gamma_K=\beta' γ K = β ′
For j ∈ { 0 , 1 } j\in\{0,1\} j ∈ { 0 , 1 } j j j ( X , A , Y ) (X,A,Y) ( X , A , Y ) ( X , R , R Y ) (X,R,RY) ( X , R , R Y ) R = 1 { A = j } R=\mathbb{1}_{\{A=j\}} R = 1 { A = j } P P P p p p w j w_j w j b j b_j b j μ j ( P ) \mu_j(P) μ j ( P ) K j K_j K j ( X , R , R Y ) (X,R,RY) ( X , R , R Y ) ( X , A , Y ′ ) (X,A,Y') ( X , A , Y ′ )
A = R if j = 1 , A = 1 − R if j = 0 , Y ′ = R Y + ( 1 − R ) ξ , A = R \text{ if } j=1,\qquad A = 1-R \text{ if } j=0,\qquad Y' = RY + (1-R)\xi, A = R if j = 1 , A = 1 − R if j = 0 , Y ′ = R Y + ( 1 − R ) ξ , where ξ ∼ Bernoulli ( 1 / 2 ) \xi\sim\operatorname{Bernoulli}(1/2) ξ ∼ Bernoulli ( 1/2 ) p p p w w w b b b p p p P ( A = j ∣ X ) = w P(A=j\mid X)=w P ( A = j ∣ X ) = w P ( A = 1 − j ∣ X ) = 1 − w P(A=1-j\mid X)=1-w P ( A = 1 − j ∣ X ) = 1 − w E [ Y ′ ∣ A = j , X ] = b \mathbb{E}[Y'\mid A=j,X]=b E [ Y ′ ∣ A = j , X ] = b E [ Y ′ ∣ A = 1 − j , X ] = 1 / 2 \mathbb{E}[Y'\mid A=1-j,X]=1/2 E [ Y ′ ∣ A = 1 − j , X ] = 1/2 μ j ( K j P ) \mu_j(K_jP) μ j ( K j P ) P P P μ 1 − j ( K j P ) = 1 / 2 \mu_{1-j}(K_jP)=1/2 μ 1 − j ( K j P ) = 1/2
μ j ( K j P ) = 1 2 + ( 2 j − 1 ) TATE ( K j P ) . (95) \mu_j(K_jP)=\frac{1}{2}+(2j-1)\operatorname{TATE}(K_jP). \tag*{(95)} μ j ( K j P ) = 2 1 + ( 2 j − 1 ) TATE ( K j P ) . (95) If w w w b b b p = 1 p=1 p = 1 K j P K_jP K j P
TATE ( K j P ) = TATT ( K j P ) = TATU ( K j P ) = ( 2 j − 1 ) ( ψ ( P ) − 1 2 ) . (96) \operatorname{TATE}(K_jP)=\operatorname{TATT}(K_jP)=\operatorname{TATU}(K_jP)=(2j-1)\left(\psi(P)-\frac{1}{2}\right). \tag*{(96)} TATE ( K j P ) = TATT ( K j P ) = TATU ( K j P ) = ( 2 j − 1 ) ( ψ ( P ) − 2 1 ) . (96) On every law of treatment observations, we write p A = E P A p_A=\mathbb{E}_P A p A = E P A E P Y − μ 0 = ∫ w ( b 1 − b 0 ) p = p A TATT \mathbb{E}_P Y-\mu_0=\int w(b_1-b_0)p=p_A\operatorname{TATT} E P Y − μ 0 = ∫ w ( b 1 − b 0 ) p = p A TATT
TATT = E P Y − μ 0 p A , μ 0 = E P Y − p A TATT , TATU = μ 1 − E P Y 1 − p A , μ 1 = E P Y + ( 1 − p A ) TATU . (97) \begin{aligned}
\operatorname{TATT}&=\frac{\mathbb{E}_P Y-\mu_0}{p_A},\qquad\mu_0=\mathbb{E}_P Y-p_A\operatorname{TATT},\\
\operatorname{TATU}&=\frac{\mu_1-\mathbb{E}_P Y}{1-p_A},\qquad\mu_1=\mathbb{E}_P Y+(1-p_A)\operatorname{TATU}. \tag*{(97)}
\end{aligned} TATT TATU = p A E P Y − μ 0 , μ 0 = E P Y − p A TATT , = 1 − p A μ 1 − E P Y , μ 1 = E P Y + ( 1 − p A ) TATU . (97) On T η \mathcal{T}_\eta T η T s e p \mathcal{T}^{\mathrm{sep}} T sep p A ∈ [ η , 1 − η ] p_A\in[\eta,1-\eta] p A ∈ [ η , 1 − η ] p A ∈ [ ϵ , 1 − ϵ ] p_A\in[\epsilon,1-\epsilon] p A ∈ [ ϵ , 1 − ϵ ] h ( O ) = ( Y , A ) h(O)=(Y,A) h ( O ) = ( Y , A ) R = [ 0 , 1 ] × [ η , 1 − η ] \mathcal{R}=[0,1]\times[\eta,1-\eta] R = [ 0 , 1 ] × [ η , 1 − η ] [ 0 , 1 ] × [ ϵ , 1 − ϵ ] [0,1]\times[\epsilon,1-\epsilon] [ 0 , 1 ] × [ ϵ , 1 − ϵ ] [ 0 , 1 ] × R [0,1]\times\mathcal{R} [ 0 , 1 ] × R μ j \mu_j μ j ( E P Y , p A ) (\mathbb{E}_P Y,p_A) ( E P Y , p A ) [ − 1 , 1 ] × R [-1,1]\times\mathcal{R} [ − 1 , 1 ] × R ( E P Y , p A ) (\mathbb{E}_P Y,p_A) ( E P Y , p A )
A fixed-accuracy pilot
The model M s e p \mathcal{M}^{\mathrm{sep}} M sep g = w p g=wp g = w p ϵ p − \epsilon p_- ϵ p − ( 1 − ϵ ) p + (1-\epsilon)p_+ ( 1 − ϵ ) p + w ^ \widehat{w} w w w w [48 ] and the cross-fitting of [45 ] . Let W = { f ∈ H α ( H 0 ) : ϵ ≤ f ≤ 1 − ϵ } \mathcal{W}=\{f\in\mathcal{H}^{\alpha}(H_0):\epsilon\leq f\leq1-\epsilon\} W = { f ∈ H α ( H 0 ) : ϵ ≤ f ≤ 1 − ϵ }
Lemma B.1 (A fixed-accuracy pilot). Fix d , α , H 0 d,\alpha,H_0 d , α , H 0 0 < ϵ < 1 / 2 0<\epsilon<1/2 0 < ϵ < 1/2 p − > 0 p_->0 p − > 0 ζ 0 ∈ ( 0 , 1 ) \zeta_0\in(0,1) ζ 0 ∈ ( 0 , 1 ) H 1 < ∞ H_1<\infty H 1 < ∞ c > 0 c>0 c > 0 n 1 ≥ 1 n_1\geq1 n 1 ≥ 1 w ^ \widehat{w} w n 1 n_1 n 1 ( X , R ) (X,R) ( X , R ) C ∞ C^\infty C ∞ [ 0 , 1 ] [0,1] [ 0 , 1 ] ( x , data ) ↦ w ^ ( x ) (x,\operatorname{data})\mapsto\widehat{w}(x) ( x , data ) ↦ w ( x ) ϵ ≤ w ^ ≤ 1 − ϵ \epsilon\leq\widehat{w}\leq1-\epsilon ϵ ≤ w ≤ 1 − ϵ ∥ w ^ ∥ C α ≤ H 1 \|\widehat{w}\|_{C^\alpha}\leq H_1 ∥ w ∥ C α ≤ H 1
P P { ∥ w ^ − w ∥ ∞ > ζ 0 } ≤ c − 1 e − c n 1 \mathbb{P}_P\{\|\widehat{w}-w\|_\infty>\zeta_0\}\leq c^{-1}e^{-cn_1} P P { ∥ w − w ∥ ∞ > ζ 0 } ≤ c − 1 e − c n 1 for every law of ( X , R ) (X,R) ( X , R ) R ∈ { 0 , 1 } R\in\{0,1\} R ∈ { 0 , 1 } p − p_- p − w = E [ R ∣ X ] ∈ W w=\mathbb{E}[R\mid X]\in\mathcal{W} w = E [ R ∣ X ] ∈ W
Proof. Let γ = min ( α , 1 ) \gamma=\min(\alpha,1) γ = min ( α , 1 ) C 0 = d H 0 C_0=\sqrt{d}H_0 C 0 = d H 0 f ∈ W f\in\mathcal{W} f ∈ W ∣ f ( x ) − f ( y ) ∣ ≤ C 0 ∥ x − y ∥ 2 γ |f(x)-f(y)|\le C_0\lVert x-y\rVert_2^\gamma ∣ f ( x ) − f ( y ) ∣ ≤ C 0 ∥ x − y ∥ 2 γ α ≤ 1 \alpha\le1 α ≤ 1 α > 1 \alpha>1 α > 1 k k k C 0 ( 2 d / k ) γ ≤ ζ 0 / 4 C_0(2\sqrt{d}/k)^\gamma\le\zeta_0/4 C 0 ( 2 d / k ) γ ≤ ζ 0 /4 [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d k d k^d k d Q Q Q Q + Q^+ Q + [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d ℓ ∞ \ell^\infty ℓ ∞ 1 / ( 2 k ) 1/(2k) 1/ ( 2 k ) Q Q Q diam Q + ≤ 2 d / k \operatorname{diam} Q^+\le2\sqrt{d}/k diam Q + ≤ 2 d / k C ∞ C^\infty C ∞ ( φ Q ) (\varphi_Q) ( φ Q ) [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d φ Q ≥ 0 \varphi_Q\ge0 φ Q ≥ 0 supp φ Q ⊆ Q + \operatorname{supp}\varphi_Q\subseteq Q^+ supp φ Q ⊆ Q + N Q N_Q N Q X i ∈ Q X_i\in Q X i ∈ Q w ~ Q \widetilde{w}_Q w Q R i R_i R i w ~ Q = 1 / 2 \widetilde{w}_Q=1/2 w Q = 1/2 N Q = 0 N_Q=0 N Q = 0 w ^ Q \widehat{w}_Q w Q w ~ Q \widetilde{w}_Q w Q [ ϵ , 1 − ϵ ] [\epsilon,1-\epsilon] [ ϵ , 1 − ϵ ] w ^ = ∑ Q w ^ Q φ Q \widehat{w}=\sum_Q\widehat{w}_Q\varphi_Q w = ∑ Q w Q φ Q w ^ \widehat{w} w C ∞ C^\infty C ∞ [ ϵ , 1 − ϵ ] [\epsilon,1-\epsilon] [ ϵ , 1 − ϵ ] ∥ w ^ ∥ C α ≤ H 1 = ∑ Q ∥ φ Q ∥ C α \lVert\widehat{w}\rVert_{C^\alpha}\le H_1=\sum_Q\lVert\varphi_Q\rVert_{C^\alpha} ∥ w ∥ C α ≤ H 1 = ∑ Q ∥ φ Q ∥ C α k k k d d d α \alpha α
Let π Q = P ( X ∈ Q ) ≥ p − k − d \pi_Q=\mathbb{P}(X\in Q)\ge p^{-k^{-d}} π Q = P ( X ∈ Q ) ≥ p − k − d m Q = E [ R ∣ X ∈ Q ] m_Q=\mathbb{E}[R\mid X\in Q] m Q = E [ R ∣ X ∈ Q ] m Q m_Q m Q w w w Q Q Q m Q ∈ [ ϵ , 1 − ϵ ] m_Q\in[\epsilon,1-\epsilon] m Q ∈ [ ϵ , 1 − ϵ ] ∣ m Q − w ( x ) ∣ ≤ ζ 0 / 4 |m_Q-w(x)|\le\zeta_0/4 ∣ m Q − w ( x ) ∣ ≤ ζ 0 /4 x ∈ Q + x\in Q^+ x ∈ Q + ∣ w ~ Q − m Q ∣ |\widetilde{w}_Q-m_Q| ∣ w Q − m Q ∣ φ Q ( x ) = 0 \varphi_Q(x)=0 φ Q ( x ) = 0 x ∈ Q + x\in Q^+ x ∈ Q +
∥ w ^ − w ∥ ∞ ≤ max Q ∣ w ~ Q − m Q ∣ + ζ 0 / 4. \lVert\widehat{w}-w\rVert_\infty\le\max_Q|\widetilde{w}_Q-m_Q|+\zeta_0/4. ∥ w − w ∥ ∞ ≤ Q max ∣ w Q − m Q ∣ + ζ 0 /4. Conditionally on N Q = s ≥ 1 N_Q=s\ge1 N Q = s ≥ 1 R i R_i R i X i ∈ Q X_i\in Q X i ∈ Q Binomial ( s , m Q ) \operatorname{Binomial}(s,m_Q) Binomial ( s , m Q ) [20 Theorem 1] implies that P { ∣ w ~ Q − m Q ∣ > ζ 0 / 2 ∣ N Q = s } ≤ 2 e − s ζ 0 2 / 2 \mathbb{P}\{|\widetilde{w}_Q-m_Q|>\zeta_0/2\mid N_Q=s\}\le2e^{-s\zeta_0^2/2} P { ∣ w Q − m Q ∣ > ζ 0 /2 ∣ N Q = s } ≤ 2 e − s ζ 0 2 /2 s = 0 s=0 s = 0 N Q ∼ Binomial ( n 1 , π Q ) N_Q\sim\operatorname{Binomial}(n_1,\pi_Q) N Q ∼ Binomial ( n 1 , π Q )
P { ∣ w ~ Q − m Q ∣ > ζ 0 / 2 } ≤ 2 ( 1 − π Q + π Q e − ζ 0 2 / 2 ) n 1 ≤ 2 e − n 1 π Q ( 1 − e − ζ 0 2 / 2 ) . \mathbb{P}\{|\widetilde{w}_Q-m_Q|>\zeta_0/2\}\le2(1-\pi_Q+\pi_Qe^{-\zeta_0^2/2})^{n_1}\le2e^{-n_1\pi_Q(1-e^{-\zeta_0^2/2})}. P { ∣ w Q − m Q ∣ > ζ 0 /2 } ≤ 2 ( 1 − π Q + π Q e − ζ 0 2 /2 ) n 1 ≤ 2 e − n 1 π Q ( 1 − e − ζ 0 2 /2 ) . A union bound over the k d k^d k d P { ∥ w ^ − w ∥ ∞ > ζ 0 } ≤ 2 k d e − n 1 p − k − d ( 1 − e − ζ 0 2 / 2 ) \mathbb{P}\{\lVert\widehat{w}-w\rVert_\infty>\zeta_0\}\le2k^d e^{-n_1p^{-k^{-d}}(1-e^{-\zeta_0^2/2})} P {∥ w − w ∥ ∞ > ζ 0 } ≤ 2 k d e − n 1 p − k − d ( 1 − e − ζ 0 2 /2 ) c − 1 e − c n 1 c^{-1}e^{-cn_1} c − 1 e − c n 1 c > 0 c>0 c > 0
**Proof of the upper bracket on M s e p . ∗ ∗ W e f i x \mathcal{M}^{\mathrm{sep}}.** We fix M sep . ∗ ∗ W e f i x θ = ( α + β ) / d \theta=(\alpha+\beta)/d θ = ( α + β ) / d ν = q α + q β \nu=q_\alpha+q_\beta ν = q α + q β ζ 0 = ζ ϵ \zeta_0=\zeta\epsilon ζ 0 = ζ ϵ n 1 = ⌊ n / 2 ⌋ n_1=\lfloor n/2\rfloor n 1 = ⌊ n /2 ⌋ N = n − n 1 N=n-n_1 N = n − n 1 P P P
D ′ = R w ^ ( X ) , U ′ = 1 , V ′ = R Y w ^ ( X ) , W ′ = 0 , λ ′ = 1 , D'=\frac{R}{\widehat{w}(X)},\qquad U'=1,\qquad V'=\frac{RY}{\widehat{w}(X)},\qquad W'=0,\qquad\lambda'=1, D ′ = w ( X ) R , U ′ = 1 , V ′ = w ( X ) R Y , W ′ = 0 , λ ′ = 1 , which depend on X X X M 0 ′ = ϵ − 1 M'_0=\epsilon^{-1} M 0 ′ = ϵ − 1
w ′ = w w ^ ≥ ϵ 1 − ϵ , a ′ = w ^ w ′ , b ′ = b , g ′ = p w w ^ , ∫ a ′ b ′ g ′ = ∫ b p . w'=\frac{w}{\widehat{w}}\ge\frac{\epsilon}{1-\epsilon},\qquad a'=\frac{\widehat{w}}{w'},\qquad b'=b,\qquad g'=\frac{pw}{\widehat{w}},\qquad\int a'b'g'=\int bp. w ′ = w w ≥ 1 − ϵ ϵ , a ′ = w ′ w , b ′ = b , g ′ = w pw , ∫ a ′ b ′ g ′ = ∫ b p . Since w ^ ∈ H α ( H 1 ) \widehat{w}\in\mathcal{H}^{\alpha}(H_1) w ∈ H α ( H 1 ) w ∈ W w\in\mathcal{W} w ∈ W a ′ = w ^ ⋅ ( 1 / w ) a'=\widehat{w}\cdot(1/w) a ′ = w ⋅ ( 1/ w ) H α ( H 2 ) \mathcal{H}^{\alpha}(H_2) H α ( H 2 ) H 2 H_2 H 2 d , α , H 0 , H 1 , ϵ d,\alpha,H_0,H_1,\epsilon d , α , H 0 , H 1 , ϵ a ′ a' a ′ ⌈ α ⌉ − 1 \lceil\alpha\rceil-1 ⌈ α ⌉ − 1 w ^ \widehat{w} w w w w 1 / w ≤ e − 1 1/w \le e^{-1} 1/ w ≤ e − 1 α − ⌈ α ⌉ + 1 \alpha-\lceil\alpha\rceil+1 α − ⌈ α ⌉ + 1 A = { ∥ w ^ − w ∥ ∞ ≤ ζ ϵ } \mathcal{A}=\{\|\widehat{w}-w\|_{\infty}\le\zeta\epsilon\} A = { ∥ w − w ∥ ∞ ≤ ζ ϵ } ∣ w / w ^ − 1 ∣ ≤ ζ |w/\widehat{w}-1|\le\zeta ∣ w / w − 1∣ ≤ ζ g ′ ∈ I ζ g'\in I_{\zeta} g ′ ∈ I ζ
Conditionally on a pilot outcome in A \mathcal{A} A P ( α , β ) \mathcal{P}(\alpha,\beta) P ( α , β ) ( D ′ , U ′ , V ′ , W ′ , λ ′ ) (D',U',V',W',\lambda') ( D ′ , U ′ , V ′ , W ′ , λ ′ ) M 0 ′ = 1 M'_0=1 M 0 ′ = 1 δ ′ = ϵ / ( 1 − ϵ ) \delta'=\epsilon/(1-\epsilon) δ ′ = ϵ / ( 1 − ϵ ) H ′ = max ( H 0 , H 2 ) H'=\max(H_0,H_2) H ′ = max ( H 0 , H 2 ) [ g − , g + ] = I ζ [g_-,g_+]=I_{\zeta} [ g − , g + ] = I ζ ψ ^ \widehat{\psi} ψ Theorem 2.3 (a) from § 5.4 , applied with these constants and w ^ \widehat{w} w N N N [ 0 , 1 ] [0,1] [ 0 , 1 ] ψ ^ \widehat{\psi} ψ A \mathcal{A} A C a ‾ N ( θ , τ I ζ , ν ) 2 C\overline{a}_N(\theta,\tau_{I_{\zeta}},\nu)^2 C a N ( θ , τ I ζ , ν ) 2 C C C A \mathcal{A} A C a ‾ N ( θ , τ I ζ , ν ) 2 + c − 1 e − c n 1 C\overline{a}_N(\theta,\tau_{I_{\zeta}},\nu)^2+c^{-1}e^{-cn_1} C a N ( θ , τ I ζ , ν ) 2 + c − 1 e − c n 1 n / 2 ≤ N ≤ n n/2\le N\le n n /2 ≤ N ≤ n N − θ N^{-\theta} N − θ log N \log N log N e − κ log N e^{-\kappa\sqrt{\log N}} e − κ l o g N n n n log n − log N ≤ log 2 / log N \sqrt{\log n}-\sqrt{\log N}\le\log2/\sqrt{\log N} log n − log N ≤ log 2/ log N a ‾ N ≤ C ′ a ‾ n \overline{a}_N\le C'\overline{a}_n a N ≤ C ′ a n e − c n 1 = o ( n − 1 ) e^{-cn_1}=o(n^{-1}) e − c n 1 = o ( n − 1 ) n − 1 / 2 ≤ a ‾ n n^{-1/2}\le\overline{a}_n n − 1/2 ≤ a n n n n r n ( ψ , M s e p ) ≤ C ζ a ‾ n ( θ , τ I ζ , ν ) r_n(\psi,\mathcal{M}^{\mathrm{sep}})\le C_{\zeta}\overline{a}_n(\theta,\tau_{I_{\zeta}},\nu) r n ( ψ , M sep ) ≤ C ζ a n ( θ , τ I ζ , ν )
Proof of Corollary 3.2 for these targets
Upper bounds. We take the choices of § 4.1 , with the response space { ( 0 , 0 ) } ∪ ( { 1 } × [ 0 , 1 ] ) \{(0,0)\}\cup(\{1\}\times[0,1]) {( 0 , 0 )} ∪ ({ 1 } × [ 0 , 1 ]) M 0 = 1 M_0=1 M 0 = 1 δ = η \delta=\eta δ = η [ g − , g + ] = [ η , η − 1 ] [g_-,g_+]=[\eta,\eta^{-1}] [ g − , g + ] = [ η , η − 1 ] M η ( α , β , H ) ⊂ P ( α , β ) \mathcal{M}_{\eta}(\alpha,\beta,H)\subset\mathcal{P}(\alpha,\beta) M η ( α , β , H ) ⊂ P ( α , β ) Theorem 2.3 (a) implies the upper bracket for the MAR mean on M η \mathcal{M}_{\eta} M η M s e p \mathcal{M}^{\mathrm{sep}} M sep I ζ I_{\zeta} I ζ H 0 > 1 H_0>1 H 0 > 1 § 3 . The map that keeps arm j j j T η ( α , β 0 , β 1 , H ) \mathcal{T}_{\eta}(\alpha,\beta_0,\beta_1,H) T η ( α , β 0 , β 1 , H ) M η ( α , β j , H ) \mathcal{M}_{\eta}(\alpha,\beta_j,H) M η ( α , β j , H ) w j w_j w j η ≤ w j ≤ 1 − η \eta\le w_j\le1-\eta η ≤ w j ≤ 1 − η 1 / w j = a j ∈ H α ( H ) 1/w_j=a_j\in\mathcal{H}^{\alpha}(H) 1/ w j = a j ∈ H α ( H ) η ≤ w j p = g j ≤ η − 1 \eta\le w_jp=g_j\le\eta^{-1} η ≤ w j p = g j ≤ η − 1 b j ∈ H β j ( H ) b_j\in\mathcal{H}^{\beta_j}(H) b j ∈ H β j ( H ) T s e p \mathcal{T}^{\mathrm{sep}} T sep M s e p \mathcal{M}^{\mathrm{sep}} M sep ( α , β j ) (\alpha,\beta_j) ( α , β j ) H 0 + 1 H_0+1 H 0 + 1 ϵ ≤ 1 − w ≤ 1 − ϵ \epsilon\le1-w\le1-\epsilon ϵ ≤ 1 − w ≤ 1 − ϵ ∥ 1 − w ∥ C α ≤ 1 + ∥ w ∥ C α \|1-w\|_{C^{\alpha}}\le1+\|w\|_{C^{\alpha}} ∥1 − w ∥ C α ≤ 1 + ∥ w ∥ C α μ j \mu_j μ j Lemma A.3 (c) shows that μ j \mu_j μ j ( ( α + β j ) / d , I , q α + q β j ) ((\alpha+\beta_j)/d,I,q_{\alpha}+q_{\beta_j}) (( α + β j ) / d , I , q α + q β j ) T η \mathcal{T}_{\eta} T η I = [ η , η − 1 ] I=[\eta,\eta^{-1}] I = [ η , η − 1 ] T s e p \mathcal{T}^{\mathrm{sep}} T sep I ζ I_{\zeta} I ζ Lemma A.3 (a), applied to T A T E = μ 1 − μ 0 \mathrm{TATE}=\mu_1-\mu_0 TATE = μ 1 − μ 0 Remark A.4 , gives the upper brackets of the three effects with the indices (24) and ν = q α + q β ∗ \nu=q_{\alpha}+q_{\beta_*} ν = q α + q β ∗
Lower bounds. We treat two cases at once. In the first, M = M η \mathcal{M}=\mathcal{M}_{\eta} M = M η T = T η \mathcal{T}=\mathcal{T}_{\eta} T = T η I = [ η , η − 1 ] I=[\eta,\eta^{-1}] I = [ η , η − 1 ] [ r − , r + ] = [ 2 η , 2 η − 1 ] [r_-,r_+]=[2\eta,2\eta^{-1}] [ r − , r + ] = [ 2 η , 2 η − 1 ] M = M s e p \mathcal{M}=\mathcal{M}^{\mathrm{sep}} M = M sep T = T s e p \mathcal{T}=\mathcal{T}^{\mathrm{sep}} T = T sep I = [ r − , r + ] = [ p − , p + ] I=[r_-,r_+]=[p_-,p_+] I = [ r − , r + ] = [ p − , p + ] r − < 1 < r + r_-<1<r_+ r − < 1 < r + η < 1 / 4 \eta<1/4 η < 1/4 ρ \rho ρ r + / r − r_+/r_- r + / r − τ \tau τ [ r − , r + ] [r_-,r_+] [ r − , r + ] τ I \tau_I τ I β ′ = β \beta'=\beta β ′ = β j = 1 j=1 j = 1 β ′ = β j \beta'=\beta_j β ′ = β j β j = min ( β 0 , β 1 ) \beta_j=\min(\beta_0,\beta_1) β j = min ( β 0 , β 1 ) j = 0 j=0 j = 0 j = 1 j=1 j = 1 s = ( α + β ′ ) / 2 s=(\alpha+\beta')/2 s = ( α + β ′ ) /2 P P P Y ∈ { 0 , 1 } Y\in\{0,1\} Y ∈ { 0 , 1 } K j P K_jP K j P w w w 1 − w 1-w 1 − w b b b j j j 1 / 2 1/2 1/2 { 0 , 1 } \{0,1\} { 0 , 1 } p p p 1 / 2 1/2 1/2 H 0 H_0 H 0 H H H P P P M \mathcal{M} M β ′ \beta' β ′ β \beta β K j P K_jP K j P T \mathcal{T} T 1 / w , 1 / ( 1 − w ) ∈ H α ( H ) 1/w,1/(1-w)\in\mathcal{H}^{\alpha}(H) 1/ w , 1/ ( 1 − w ) ∈ H α ( H ) b ∈ H β ′ ( H ) b\in\mathcal{H}^{\beta'}(H) b ∈ H β ′ ( H ) η ≤ w ≤ 1 − η \eta\le w\le1-\eta η ≤ w ≤ 1 − η η ≤ w p , ( 1 − w ) p ≤ η − 1 \eta\le wp,(1-w)p\le\eta^{-1} η ≤ w p , ( 1 − w ) p ≤ η − 1 w , 1 − w ∈ H α ( H 0 ) w,1-w\in\mathcal{H}^{\alpha}(H_0) w , 1 − w ∈ H α ( H 0 ) b ∈ H β ′ ( H 0 ) b\in\mathcal{H}^{\beta'}(H_0) b ∈ H β ′ ( H 0 ) ϵ ≤ w ≤ 1 − ϵ \epsilon\le w \le1-\epsilon ϵ ≤ w ≤ 1 − ϵ p − ≤ p ≤ p + p_- \le p \le p_+ p − ≤ p ≤ p + Lemma 7.2 , for functions K K K k 0 = 2 k_0=2 k 0 = 2 k 0 = 1 / 2 k_0=1/2 k 0 = 1/2 K ≡ 1 K\equiv1 K ≡ 1 2 < H 2 < H 2 < H 1 / 2 < H 0 1/2 < H_0 1/2 < H 0 η < 1 / 2 < 1 − η \eta< 1/2 < 1-\eta η < 1/2 < 1 − η ϵ < 1 / 2 < 1 − ϵ \epsilon< 1/2 < 1-\epsilon ϵ < 1/2 < 1 − ϵ r ± / 2 = η ∓ 1 r_\pm/2=\eta^{\mp1} r ± /2 = η ∓ 1 r ± = p ± r_\pm=p_\pm r ± = p ±
Suppose first that s < d / 4 s<d/4 s < d /4 Lemma 7.2 , applied with a small enough fixed value of the input ϵ \epsilon ϵ Proposition 7.1 (not the overlap bound of T sep \mathcal{T}^{\operatorname{sep}} T sep ( α , β ′ ) (\alpha,\beta') ( α , β ′ ) [ r − , r + ] [r_-,r_+] [ r − , r + ] M \mathcal{M} M P n ′ = K j P n h \mathcal{P}'_n=K_j\mathcal{P}^h_n P n ′ = K j P n h T \mathcal{T} T n n n θ = 2 s / d \theta=2s/d θ = 2 s / d t n = c 1 a n ( θ , τ I ) t_n=c_1a_n(\theta,\tau_I) t n = c 1 a n ( θ , τ I ) Lemma A.2 gives the lower bracket on M \mathcal{M} M Lemma A.3 (c) shows that μ j \mu_j μ j t n t_n t n ( P n ′ ) (\mathcal{P}'_n) ( P n ′ ) T A T E T_{\mathrm{ATE}} T ATE μ j \mu_j μ j P n ′ \mathcal{P}'_n P n ′ t n t_n t n Lemma A.3 (b), with h ( O ) = ( Y , A ) h(O)=(Y,A) h ( O ) = ( Y , A ) μ 0 \mu_0 μ 0 μ 1 \mu_1 μ 1 n t n 2 → ∞ nt_n^2\to\infty n t n 2 → ∞
log ( n t n 2 ) = ( 1 − 2 θ ) log n − 2 κ log n + ( θ + 2 ) log log n + 2 log c 1 , \log(nt_n^2)=(1-2\theta)\log n-2\kappa\sqrt{\log n}+(\theta+2)\log\log n+2\log c_1, log ( n t n 2 ) = ( 1 − 2 θ ) log n − 2 κ log n + ( θ + 2 ) log log n + 2 log c 1 , so that the target is hard at scale t n / ( 4 L ) t_n/(4L) t n / ( 4 L ) ( P n ′ ) (\mathcal{P}'_n) ( P n ′ ) T \mathcal{T} T Lemma A.2 .
If s ≥ d / 4 s\ge d/4 s ≥ d /4 Q t Q_t Q t X X X [ 0 , 1 ] d [0,1]^d [ 0 , 1 ] d Z ∼ π u , v Z\sim\pi_{u,v} Z ∼ π u , v ( u , v ) = ( 0 , t / 2 ) (u,v)=(0,t/2) ( u , v ) = ( 0 , t /2 ) ∣ t ∣ < 1 / 8 |t|<1/8 ∣ t ∣ < 1/8 p = 1 p=1 p = 1 w = 1 / 2 w=1/2 w = 1/2 b = 1 / 2 + t b=1/2+t b = 1/2 + t Q t ∈ M Q_t\in\mathcal{M} Q t ∈ M K j Q t ∈ T K_jQ_t\in\mathcal{T} K j Q t ∈ T ψ ( Q t ) = 1 / 2 + t \psi(Q_t)=1/2+t ψ ( Q t ) = 1/2 + t Q t Q_t Q t Q 0 Q_0 Q 0 1 + t s v ( Z ) / 2 1+ts_v(Z)/2 1 + t s v ( Z ) /2 t t t 3 / 4 3/4 3/4 Lemma 5.1 implies that r n ( ψ , M ) ≥ c n − 1 / 2 r_n(\psi,\mathcal{M})\ge cn^{-1/2} r n ( ψ , M ) ≥ c n − 1/2 T T T K j Q t K_jQ_t K j Q t ( 2 j − 1 ) t (2j-1)t ( 2 j − 1 ) t Lemma A.3 (c), applied to the target ( 2 j − 1 ) T + 1 / 2 (2j-1)T+1/2 ( 2 j − 1 ) T + 1/2 T \mathcal{T} T r n ( T , T ) ≥ c n − 1 / 2 r_n(T,\mathcal{T})\ge cn^{-1/2} r n ( T , T ) ≥ c n − 1/2
For the MAR mean on M η \mathcal{M}_\eta M η P r ( π 0 ) \mathcal{P}_r(\pi_0) P r ( π 0 ) M η \mathcal{M}_\eta M η Theorem 2.3 (b). The baseline π 0 \pi_0 π 0 (8) : Σ \Sigma Σ 0 < η < 1 / 4 0<\eta<1/4 0 < η < 1/4 H > 2 H>2 H > 2 max { δ , g − } = η < w 0 = 1 / 2 < η − 1 = g + \max\{\delta,g_-\}=\eta<w_0=1/2<\eta^{-1}=g_+ max { δ , g − } = η < w 0 = 1/2 < η − 1 = g + ∣ a 0 ∣ = 2 < H |a_0|=2<H ∣ a 0 ∣ = 2 < H ∣ b 0 ∣ = 1 / 2 < H |b_0|=1/2<H ∣ b 0 ∣ = 1/2 < H P r ( π 0 ) ⊆ M η \mathcal{P}_r(\pi_0)\subseteq\mathcal{M}_\eta P r ( π 0 ) ⊆ M η 0 < r < 1 / 2 − η 0<r<1/2-\eta 0 < r < 1/2 − η w ≤ 1 / 2 + r < 1 − η w\le1/2+r<1-\eta w ≤ 1/2 + r < 1 − η P r ( π 0 ) \mathcal{P}_r(\pi_0) P r ( π 0 )
The logarithmic asymptotics. Let P \mathcal{P} P M sep \mathcal{M}^{\operatorname{sep}} M sep T sep \mathcal{T}^{\operatorname{sep}} T sep s < d / 4 s<d/4 s < d /4 θ = 2 s / d \theta=2s/d θ = 2 s / d κ ζ \kappa_\zeta κ ζ κ \kappa κ (12) computed from τ I ζ \tau_{I_\zeta} τ I ζ log a n ( θ , τ ) = − θ log n − κ log n + ( θ / 2 ) log log n \log a_n(\theta,\tau)=-\theta\log n-\kappa\sqrt{\log n}+(\theta/2)\log\log n log a n ( θ , τ ) = − θ log n − κ log n + ( θ /2 ) log log n [ p − , p + ] [p_-,p_+] [ p − , p + ] I ζ I_\zeta I ζ
− κ p ( θ ) ≤ lim inf n → ∞ log r n + θ log n log n ≤ lim sup n → ∞ log r n + θ log n log n ≤ − κ ζ . -\kappa_p(\theta)\le\liminf_{n\to\infty}\frac{\log r_n+\theta\log n}{\sqrt{\log n}}\le\limsup_{n\to\infty}\frac{\log r_n+\theta\log n}{\sqrt{\log n}}\le-\kappa_\zeta. − κ p ( θ ) ≤ n → ∞ lim inf log n log r n + θ log n ≤ n → ∞ lim sup log n log r n + θ log n ≤ − κ ζ . As ζ → 0 \zeta\to0 ζ → 0 ρ I ζ → ρ [ p − , p + ] \rho_{I_\zeta}\to\rho_{[p_-,p_+]} ρ I ζ → ρ [ p − , p + ] τ I ζ → τ p \tau_{I_\zeta}\to\tau_p τ I ζ → τ p κ ζ → κ p ( θ ) \kappa_\zeta\to\kappa_p(\theta) κ ζ → κ p ( θ ) (21) .
Conditional covariances and variances
Bilinear and quadratic targets. We apply Theorem 2.3 with D = 1 D=1 D = 1 M 0 = 1 M_0=1 M 0 = 1 H = H 0 H=H_0 H = H 0 [ g − , g + ] = [ p − , p + ] [g_-,g_+]=[p_-,p_+] [ g − , g + ] = [ p − , p + ] δ = 1 / 2 \delta=1/2 δ = 1/2 ( W , λ ) = ( 0 , 1 ) (W,\lambda)=(0,1) ( W , λ ) = ( 0 , 1 ) E P [ m U m V ] \mathbb{E}_P[m_Um_V] E P [ m U m V ] Q 2 Q_2 Q 2 ( W , λ ) = ( U V , − 1 ) (W,\lambda)=(UV,-1) ( W , λ ) = ( U V , − 1 ) E P [ Cov P ( U , V ∣ X ) ] \mathbb{E}_P[\operatorname{Cov}_P(U,V\mid X)] E P [ Cov P ( U , V ∣ X )] σ ˉ 2 \bar{\sigma}^2 σ ˉ 2 w = 1 w=1 w = 1 a = m U a=m_U a = m U b = m V b=m_V b = m V g = p g=p g = p P b i l ( α , β ) \mathcal{P}_{\mathrm{bil}}(\alpha,\beta) P bil ( α , β ) P q u a d ( s ) \mathcal{P}_{\mathrm{quad}}(s) P quad ( s ) U = V = Y U=V=Y U = V = Y α = β = s \alpha=\beta=s α = β = s U , V U,V U , V ( w 0 , a 0 , b 0 ) = ( 1 , 0 , 0 ) (w_0,a_0,b_0)=(1,0,0) ( w 0 , a 0 , b 0 ) = ( 1 , 0 , 0 ) (8) is the identity, and the strict inequalities there follow from (20) and H 0 > 1 H_0>1 H 0 > 1 Y Y Y
Upper brackets. We write m j = E P [ Y j ] m_j=\mathbb{E}_P[Y^j] m j = E P [ Y j ] j = 1 , 2 j=1,2 j = 1 , 2 f 0 f_0 f 0 B 0 = ∥ f 0 ∥ ∞ < ∞ B_0=\lVert f_0\rVert_\infty<\infty B 0 = ∥ f 0 ∥ ∞ < ∞ ℓ 0 = E P [ f 0 ( X ) 2 − 2 Y f 0 ( X ) ] \ell_0=\mathbb{E}_P[f_0(X)^2-2Yf_0(X)] ℓ 0 = E P [ f 0 ( X ) 2 − 2 Y f 0 ( X )] E f 0 = E P [ ( f ( X ) − f 0 ( X ) ) 2 ] E_{f_0}=\mathbb{E}_P[(f(X)-f_0(X))^2] E f 0 = E P [( f ( X ) − f 0 ( X ) ) 2 ] E P [ Y f 0 ( X ) ] = E P [ f ( X ) f 0 ( X ) ] \mathbb{E}_P[Yf_0(X)]=\mathbb{E}_P[f(X)f_0(X)] E P [ Y f 0 ( X )] = E P [ f ( X ) f 0 ( X )]
E ∗ = Q 2 − m 1 2 , R ∗ 2 = Q 2 − m 1 2 max { m 2 − m 1 2 , v min } , E f 0 = Q 2 + ℓ 0 , E_* = Q_2-m_1^2,\qquad R_*^2=\frac{Q_2-m_1^2}{\max\{m_2-m_1^2,v_{\min}\}},\qquad E_{f_0}=Q_2+\ell_0, E ∗ = Q 2 − m 1 2 , R ∗ 2 = max { m 2 − m 1 2 , v m i n } Q 2 − m 1 2 , E f 0 = Q 2 + ℓ 0 , where the middle identity holds on P q u a d ( s ; v min ) \mathcal{P}_{\mathrm{quad}}(s;v_{\min}) P quad ( s ; v m i n ) Lemma A.3 , we use h ( O ) = ( Y , Y 2 ) h(O)=(Y,Y^2) h ( O ) = ( Y , Y 2 ) R = [ − 1 , 1 ] × [ 0 , 1 ] \mathcal{R}=[-1,1]\times[0,1] R = [ − 1 , 1 ] × [ 0 , 1 ] E ∗ E_* E ∗ R ∗ 2 R_*^2 R ∗ 2 h ( O ) = f 0 ( X ) 2 − 2 Y f 0 ( X ) h(O)=f_0(X)^2-2Yf_0(X) h ( O ) = f 0 ( X ) 2 − 2 Y f 0 ( X ) R = [ − B 0 2 − 2 B 0 , B 0 2 + 2 B 0 ] \mathcal{R}=[-B_0^2-2B_0,B_0^2+2B_0] R = [ − B 0 2 − 2 B 0 , B 0 2 + 2 B 0 ] E f 0 E_{f_0} E f 0 v min v_{\min} v m i n [ 0 , 1 ] × R [0,1]\times\mathcal{R} [ 0 , 1 ] × R Q 2 Q_2 Q 2 Lemma A.3 (a), the three targets satisfy the upper bracket ( 2 s / d , [ p − , p + ] , 2 q s ) (2s/d,[p_-,p_+],2q_s) ( 2 s / d , [ p − , p + ] , 2 q s )
Lower brackets below the threshold. Let s < d / 4 s<d/4 s < d /4 § 10 .1 in the diagonal case, with Rademacher baseline, ( D , U , V , W , λ ) = ( 1 , Y , Y , 0 , 1 ) (D,U,V,W,\lambda)=(1,Y,Y,0,1) ( D , U , V , W , λ ) = ( 1 , Y , Y , 0 , 1 ) s d = Y s_d=Y s d = Y ( s , s ) (s,s) ( s , s ) [ r − , r + ] = [ p − , p + ] [r_-,r_+]=[p_-,p_+] [ r − , r + ] = [ p − , p + ] P ( Y = y ∣ X ) = { 1 + y ( u + v ) } / 2 P(Y=y\mid X)=\{1+y(u+v)\}/2 P ( Y = y ∣ X ) = { 1 + y ( u + v )} /2 f = u + v f=u+v f = u + v Q 2 Q_2 Q 2 P q u a d ( s ) \mathcal{P}_{\mathrm{quad}}(s) P quad ( s ) K = u + v K=u+v K = u + v k 0 = 0 k_0=0 k 0 = 0 γ K = s \gamma_K=s γ K = s K ≡ 1 K\equiv1 K ≡ 1 Lemma 7.2 , for a suitable fixed ϵ \epsilon ϵ P n h \mathcal{P}_n^h P n h P q u a d ( s ) \mathcal{P}_{\mathrm{quad}}(s) P quad ( s ) n n n Y 2 = 1 Y^2=1 Y 2 = 1 ∣ E P Y ∣ ≤ ∥ u + v ∥ ∞ ≤ C n − s / d |\mathbb{E}_P Y|\leq\lVert u+v\rVert_\infty\leq Cn^{-s/d} ∣ E P Y ∣ ≤ ∥ u + v ∥ ∞ ≤ C n − s / d P q u a d ( s ; v min ) \mathcal{P}_{\mathrm{quad}}(s;v_{\min}) P quad ( s ; v m i n ) n n n Q 2 Q_2 Q 2 t n = c 1 a n ( 2 s / d , τ p ) t_n=c_1a_n(2s/d,\tau_p) t n = c 1 a n ( 2 s / d , τ p ) n t n 2 → ∞ nt_n^2\to\infty n t n 2 → ∞ Q 2 = E ∗ + m 1 2 = ( m 2 − m 1 2 ) R ∗ 2 + m 1 2 = E f 0 − ℓ 0 Q_2=E_*+m_1^2=(m_2-m_1^2)R_*^2+m_1^2=E_{f_0}-\ell_0 Q 2 = E ∗ + m 1 2 = ( m 2 − m 1 2 ) R ∗ 2 + m 1 2 = E f 0 − ℓ 0 L L L I 1 × R I_1\times\mathcal{R} I 1 × R L = 2 , 5 , 1 L=2,\sqrt{5},1 L = 2 , 5 , 1 I 1 = [ 0 , 1 ] I_1=[0,1] I 1 = [ 0 , 1 ] E ∗ E_* E ∗ R ∗ 2 R_*^2 R ∗ 2 I 1 = [ 0 , ( 1 + B 0 ) 2 ] I_1=[0,(1+B_0)^2] I 1 = [ 0 , ( 1 + B 0 ) 2 ] E f 0 E_{f_0} E f 0 Lemma A.3 (b), with T 1 = Q 2 T_1=Q_2 T 1 = Q 2 e n = 0 e_n=0 e n = 0 ( P n h ) (\mathcal{P}_n^h) ( P n h ) t n / ( 4 L ) t_n/(4L) t n / ( 4 L ) Lemma A.2 .
Root-n n n We fix ϕ = η sin ( 2 π x 1 ) \phi=\eta\sin(2\pi x_1) ϕ = η sin ( 2 π x 1 ) η > 0 \eta>0 η > 0 s , H 0 s,H_0 s , H 0 ∥ ϕ ∥ C s ≤ H 0 \lVert\phi\rVert_{C^s}\leq H_0 ∥ ϕ ∥ C s ≤ H 0 ∥ ϕ ∥ ∞ ≤ 1 / 2 \lVert\phi\rVert_\infty\leq1/2 ∥ ϕ ∥ ∞ ≤ 1/2 P t P_t P t p = 1 p=1 p = 1 P t ( Y = y ∣ X ) = { 1 + P_t(Y=y\mid X)=\{1+ P t ( Y = y ∣ X ) = { 1 + y t φ ( X ) } / 2 yt\varphi(X)\}/2 y tφ ( X )} /2 ∣ t ∣ < 1 |t| < 1 ∣ t ∣ < 1 P q u a d ( s ; v min ) \mathcal{P}_{\mathrm{quad}}(s; v_{\min}) P quad ( s ; v m i n ) E P t Y = 0 \mathbb{E}_{P_t}Y = 0 E P t Y = 0 Var P t ( Y ) = 1 \operatorname{Var}_{P_t}(Y) = 1 Var P t ( Y ) = 1 A φ = ∫ φ 2 > 0 A_\varphi= \int\varphi^2 > 0 A φ = ∫ φ 2 > 0 B φ = ∫ φ f 0 B_\varphi= \int\varphi f_0 B φ = ∫ φ f 0 E ∗ ( P t ) = R ∗ 2 ( P t ) = t 2 A φ \mathbb{E}_*(P_t) = R_*^2(P_t) = t^2A_\varphi E ∗ ( P t ) = R ∗ 2 ( P t ) = t 2 A φ E f 0 ( P t ) = t 2 A φ − 2 t B φ + ∫ f 0 2 E_{f_0}(P_t) = t^2A_\varphi- 2tB_\varphi+ \int f_0^2 E f 0 ( P t ) = t 2 A φ − 2 t B φ + ∫ f 0 2 t = 1 / 4 t = 1/4 t = 1/4 t = 1 / 2 t = 1/2 t = 1/2 A φ / 2 A_\varphi/2 A φ /2 A φ / 4 A_\varphi/4 A φ /4 E ∗ E_* E ∗ R ∗ 2 R_*^2 R ∗ 2 A φ / 2 A_\varphi/2 A φ /2 1 + t Y φ ( X ) ≥ 1 / 2 1 + tY\varphi(X) \ge1/2 1 + t Y φ ( X ) ≥ 1/2 n n n E f 0 E_{f_0} E f 0 B 0 B_0 B 0
The trial. The map ( X , A , Y o b s ) ↦ ( X , ( 2 A − 1 ) ( 2 Y o b s − 1 ) ) (X,A,Y_{\mathrm{obs}}) \mapsto(X,(2A - 1)(2Y_{\mathrm{obs}} - 1)) ( X , A , Y obs ) ↦ ( X , ( 2 A − 1 ) ( 2 Y obs − 1 )) P q u a d ( s ) \mathcal{P}_{\mathrm{quad}}(s) P quad ( s ) [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] f f f A ∼ Bernoulli ( 1 / 2 ) A \sim\operatorname{Bernoulli}(1/2) A ∼ Bernoulli ( 1/2 ) ( X , Y ) (X,Y) ( X , Y ) ( X , A , { 1 + ( 2 A − 1 ) Y } / 2 ) (X,A,\{1 + (2A - 1)Y\}/2) ( X , A , { 1 + ( 2 A − 1 ) Y } /2 ) [ 0 , 1 ] [0,1] [ 0 , 1 ] ( 1 ± f ) / 2 (1 \pm f)/2 ( 1 ± f ) /2 P q u a d ( s ) \mathcal{P}_{\mathrm{quad}}(s) P quad ( s ) p p p f f f
Wald ratios and the overlap-weighted effect
We recall the causal interpretations of the targets of § 4.3 and prove Corollary 3.2 . The interpretations use identification assumptions; the classes restrict observed laws.
Wald ratios
Example D.1 (Wald ratio). Let A ( z ) A(z) A ( z ) Y ( a ) Y(a) Y ( a ) A = A ( Z ) A = A(Z) A = A ( Z ) Y = Y ( A ) Y = Y(A) Y = Y ( A ) Z ⊥ ( A ( 0 ) , A ( 1 ) , Y ( 0 ) , Y ( 1 ) ) ∣ X Z \perp(A(0), A(1), Y(0), Y(1)) \mid X Z ⊥ ( A ( 0 ) , A ( 1 ) , Y ( 0 ) , Y ( 1 )) ∣ X A ( 1 ) ≥ A ( 0 ) A(1) \ge A(0) A ( 1 ) ≥ A ( 0 ) T W T_{\mathrm{W}} T W E [ Y ( 1 ) − Y ( 0 ) ∣ A ( 1 ) > A ( 0 ) ] \mathbb{E}[Y(1) - Y(0) \mid A(1) > A(0)] E [ Y ( 1 ) − Y ( 0 ) ∣ A ( 1 ) > A ( 0 )] [2 , 21 ] , in the form with covariates given in [11 , 16 ] . Its numerator and denominator are average effects of the instrument, that is, differences of two MAR means.
Example D.2 (Average conditional Wald ratio). Suppose that P ( Z = 1 ∣ X ) = 1 / 2 P(Z = 1 \mid X) = 1/2 P ( Z = 1 ∣ X ) = 1/2 A ≤ Z A \le Z A ≤ Z Z = 0 Z = 0 Z = 0 b ( X ) b(X) b ( X ) E [ Y ( 1 ) − Y ( 0 ) ∣ X , A ( 1 ) ] = E [ Y ( 1 ) − Y ( 0 ) ∣ X ] \mathbb{E}[Y(1) - Y(0) \mid X, A(1)] = \mathbb{E}[Y(1) - Y(0) \mid X] E [ Y ( 1 ) − Y ( 0 ) ∣ X , A ( 1 )] = E [ Y ( 1 ) − Y ( 0 ) ∣ X ] T C W T_{\mathrm{CW}} T CW [70 Theorem 1 and the following discussion] . The two Wald targets differ in general, since T W T_{\mathrm{W}} T W b ( X ) b(X) b ( X ) q ( X ) q(X) q ( X )
Proof of Corollary 3.2 for the Wald ratios. Wald ratio. We write N = E P [ Δ Y ( X ) ] ∈ [ − 1 , 1 ] \mathcal{N} = \mathbb{E}_{P}[\Delta_Y(X)] \in[-1,1] N = E P [ Δ Y ( X )] ∈ [ − 1 , 1 ] D = E P [ Δ A ( X ) ] ∈ [ c W , 1 ] \mathcal{D} = \mathbb{E}_{P}[\Delta_A(X)] \in[c_{\mathrm{W}},1] D = E P [ Δ A ( X )] ∈ [ c W , 1 ] ( X , Z , A , Y ) ↦ ( X , Z , Y ) (X,Z,A,Y) \mapsto(X,Z,Y) ( X , Z , A , Y ) ↦ ( X , Z , Y ) ( X , Z , A , Y ) ↦ ( X , Z , A ) (X,Z,A,Y) \mapsto(X,Z,A) ( X , Z , A , Y ) ↦ ( X , Z , A ) I η , c W ( α , β , H ) \mathcal{I}_{\eta,c_{\mathrm{W}}}(\alpha,\beta,H) I η , c W ( α , β , H ) T η ( α , β , β , H ) \mathcal{T}_{\eta}(\alpha,\beta,\beta,H) T η ( α , β , β , H ) N \mathcal{N} N D \mathcal{D} D Corollary 3.2 for the ATE on T η ( α , β , β , H ) \mathcal{T}_{\eta}(\alpha,\beta,\beta,H) T η ( α , β , β , H ) Lemma A.3 (c), both satisfy the upper bracket ( ( α + β ) / d , [ η , η − 1 ] , q α + q β ) ((\alpha+\beta)/d,[\eta,\eta^{-1}],q_{\alpha}+q_{\beta}) (( α + β ) / d , [ η , η − 1 ] , q α + q β ) I η , c W ( α , β , H ) \mathcal{I}_{\eta,c_{\mathrm{W}}}(\alpha,\beta,H) I η , c W ( α , β , H ) Lemma A.3 (a) to the ratio, with C 0 = 1 C_0=1 C 0 = 1 δ D = c W \delta_{\mathcal{D}}=c_{\mathrm{W}} δ D = c W Remark A.4 , we obtain the upper bracket for T W T_{\mathrm{W}} T W ( X , Z , Y ) ↦ ( X , Z , Z , Y ) (X,Z,Y) \mapsto(X,Z,Z,Y) ( X , Z , Y ) ↦ ( X , Z , Z , Y ) A = Z A=Z A = Z T η ( α , β , β , H ) \mathcal{T}_{\eta}(\alpha,\beta,\beta,H) T η ( α , β , β , H ) I η , c W ( α , β , H ) \mathcal{I}_{\eta,c_{\mathrm{W}}}(\alpha,\beta,H) I η , c W ( α , β , H ) m A , z = z m_{A,z}=z m A , z = z z ≤ 1 < H z \le1 < H z ≤ 1 < H Δ A = 1 ≥ c W \Delta_A=1 \ge c_{\mathrm{W}} Δ A = 1 ≥ c W T η ( α , β , β , H ) \mathcal{T}_{\eta}(\alpha,\beta,\beta,H) T η ( α , β , β , H ) Z Z Z T W T_{\mathrm{W}} T W Lemma A.3 (c) transfers its lower bracket.
Average conditional Wald ratio. We take the response ( Z , A , Y ) (Z,A,Y) ( Z , A , Y ) ( D , U , V , W , λ ) = ( 2 A , 1 , 2 ( 2 Z − 1 ) Y , 0 , 1 ) (D,U,V,W,\lambda)=(2A,1,2(2Z-1)Y,0,1) ( D , U , V , W , λ ) = ( 2 A , 1 , 2 ( 2 Z − 1 ) Y , 0 , 1 ) M 0 = 2 M_0=2 M 0 = 2 P ( Z = 1 ∣ X ) = 1 / 2 P(Z=1 \mid X)=1/2 P ( Z = 1 ∣ X ) = 1/2 A ≤ Z A \le Z A ≤ Z E P [ 2 A ∣ X ] = q \mathbb{E}_{P}[2A \mid X] = q E P [ 2 A ∣ X ] = q E P [ 2 ( 2 Z − 1 ) Y ∣ X ] = Δ Y \mathbb{E}_{P}[2(2Z-1)Y \mid X] = \Delta_Y E P [ 2 ( 2 Z − 1 ) Y ∣ X ] = Δ Y w = q w=q w = q a = 1 / q a=1/q a = 1/ q b = Δ Y / q b=\Delta_Y/q b = Δ Y / q g = q p g=qp g = q p ψ = ( Δ Y / q ) p = T C W \psi= (\Delta_Y/q)p = T_{\mathrm{CW}} ψ = ( Δ Y / q ) p = T CW J η ( α , β , H ) ⊆ P ( α , β ) \mathcal{J}_{\eta}(\alpha,\beta,H) \subseteq\mathcal{P}(\alpha,\beta) J η ( α , β , H ) ⊆ P ( α , β ) δ = η \delta=\eta δ = η [ g − , g + ] = [ η , η − 1 ] [g_-,g_+] = [\eta,\eta^{-1}] [ g − , g + ] = [ η , η − 1 ] Theorem 2.3 (a). For the lower bracket, the kernel that draws Z ∼ Bernoulli ( 1 / 2 ) Z \sim\operatorname{Bernoulli}(1/2) Z ∼ Bernoulli ( 1/2 ) ( X , R , R B ) (X,R,RB) ( X , R , R B ) ( X , Z , Z R , Z R B ) (X,Z,ZR,ZRB) ( X , Z , Z R , Z R B ) M η ( α , β , H ) \mathcal{M}_{\eta}(\alpha,\beta,H) M η ( α , β , H ) J η ( α , β , H ) \mathcal{J}_{\eta}(\alpha,\beta,H) J η ( α , β , H ) A = Z R ≤ Z A=ZR \le Z A = Z R ≤ Z q = P ( R = 1 ∣ X ) q=P(R=1 \mid X) q = P ( R = 1 ∣ X ) Δ Y = q b \Delta_Y=qb Δ Y = q b b = E [ B ∣ R = 1 , X ] b=\mathbb{E}[B \mid R=1,X] b = E [ B ∣ R = 1 , X ] b b b
The overlap-weighted effect
Example D.3 (Overlap-weighted effect). For fixed γ \gamma γ E P [ ( Y − γ A − h ( X ) ) 2 ] \mathbb{E}_{P}[(Y-\gamma A-h(X))^2] E P [( Y − γ A − h ( X ) ) 2 ] h h h h = m − γ w h=m-\gamma w h = m − γ w γ \gamma γ E P [ Y ∣ A , X ] \mathbb{E}_{P}[Y \mid A,X] E P [ Y ∣ A , X ] § 4.2 . Under consistency and conditional exchangeability, (3.8) is the overlap-weighted treatment effect [35 ] ; see also [12 Remark 9.3.3] for this interpretation of regression on residuals when effects vary with X X X
The weights w ( 1 − w ) w(1-w) w ( 1 − w ) [13 Section 5.4, Corollary 5.2] , as the weights that minimize the efficiency bound for weighted sample average treatment effects under homoscedasticity.
Robins et al. [48 Example 1c] impose smoothness on m − γ O W w m-\gamma_{\mathrm{OW}}w m − γ OW w m m m ∣ γ O W ∣ ≤ 1 |\gamma_{\mathrm{OW}}| \le1 ∣ γ OW ∣ ≤ 1 H α ( H ) ⊆ H β ( C H ) \mathcal{H}^{\alpha}(H) \subseteq\mathcal{H}^{\beta}(CH) H α ( H ) ⊆ H β ( C H ) β ≤ α \beta\le\alpha β ≤ α β ≤ α \beta\le\alpha β ≤ α β > α \beta> \alpha β > α [48 Remark 4.2] refer to n − 2 s / d n^{-2s/d} n − 2 s / d s < d / 4 s<d/4 s < d /4 s s s β ≤ α \beta\le\alpha β ≤ α Corollary 3.2 confirms the exponent 2 s / d 2s/d 2 s / d s = ( α + β ) / 2 s=(\alpha+\beta)/2 s = ( α + β ) /2 n − 2 s / d n^{-2s/d} n − 2 s / d e − ( κ p ( 2 s / d ) + o ( 1 ) ) log n e^{-(\kappa_p(2s/d)+o(1))\sqrt{\log n}} e − ( κ p ( 2 s / d ) + o ( 1 )) l o g n [46 Section 2] reports minimax bounds under a known positive lower bound on the denominator, with effective smoothness min ( α + β , 2 β ) \min(\alpha+\beta,2\beta) min ( α + β , 2 β ) E [ Cov ( A , Y ∣ X ) ] / E [ Var ( Y ∣ X ) ] \mathbb{E}[\operatorname{Cov}(A,Y \mid X)]/\mathbb{E}[\operatorname{Var}(Y \mid X)] E [ Cov ( A , Y ∣ X )] / E [ Var ( Y ∣ X )] γ O W \gamma_{\mathrm{OW}} γ OW A A A Y Y Y [46 ] (Theorem S14), which was not available to us.
Proof of Corollary 3.2 for the overlap-weighted effect. Here w = P ( A = 1 ∣ X ) w=P(A=1 \mid X) w = P ( A = 1 ∣ X ) D D D γ O W = N / D \gamma_{\mathrm{OW}}=\mathcal{N}/\mathcal{D} γ OW = N / D N = E P [ Cov ( A , Y ∣ X ) ] ∈ [ − 1 / 4 , 1 / 4 ] \mathcal{N}=\mathbb{E}_{P}[\operatorname{Cov}(A,Y \mid X)] \in[-1/4,1/4] N = E P [ Cov ( A , Y ∣ X )] ∈ [ − 1/4 , 1/4 ] D = E P [ w ( 1 − w ) ] ∈ [ ϵ ( 1 − ϵ ) , 1 / 4 ] \mathcal{D}=\mathbb{E}_{P}[w(1-w)] \in[\epsilon(1-\epsilon),1/4] D = E P [ w ( 1 − w )] ∈ [ ϵ ( 1 − ϵ ) , 1/4 ] θ = min { ( α + β ) / d , 2 α / d } \theta=\min\{(\alpha+\beta)/d,2\alpha/d\} θ = min {( α + β ) / d , 2 α / d }
Upper bracket. We apply Theorem 2.3 (a) with D = 1 D=1 D = 1 M 0 = 1 M_0=1 M 0 = 1 H = H 0 H=H_0 H = H 0 δ = 1 / 2 \delta=1/2 δ = 1/2 [ g − , g + ] = [ p − , p + ] [g_-,g_+]=[p_-,p_+] [ g − , g + ] = [ p − , p + ] ( U , V , W , λ ) = ( A , Y , A Y , − 1 ) (U,V,W,\lambda)=(A,Y,AY,-1) ( U , V , W , λ ) = ( A , Y , A Y , − 1 ) ( A , A , A , − 1 ) (A,A,A,-1) ( A , A , A , − 1 ) N \mathcal{N} N D \mathcal{D} D ( ( α + β ) / d , [ p − , p + ] , q α + q β ) ((\alpha+\beta)/d,[p_-,p_+],q_\alpha+q_\beta) (( α + β ) / d , [ p − , p + ] , q α + q β ) ( 2 α / d , [ p − , p + ] , 2 q α ) (2\alpha/d,[p_-,p_+],2q_\alpha) ( 2 α / d , [ p − , p + ] , 2 q α ) Lemma A.3 (a) to the ratio with C 0 = 1 / 4 C_0=1/4 C 0 = 1/4 δ D = ϵ ( 1 − ϵ ) \delta_{\mathcal{D}}=\epsilon(1-\epsilon) δ D = ϵ ( 1 − ϵ ) Remark A.4 , we obtain the upper bracket with exponent θ \theta θ ν = q α + q min ( α , β ) \nu=q_\alpha+q_{\min(\alpha,\beta)} ν = q α + q m i n ( α , β )
A common response family. Let S 1 = 2 A − 1 S_1=2A-1 S 1 = 2 A − 1 S 2 = 2 Y − 1 S_2=2Y-1 S 2 = 2 Y − 1 c ∈ { 0 , 1 / 4 } c\in\{0,1/4\} c ∈ { 0 , 1/4 }
π 0 ( c ) = 1 + c S 1 S 2 4 , s i = 2 S i 1 + c S 1 S 2 ( i = 1 , 2 ) . \pi_0^{(c)}=\frac{1+cS_1S_2}{4},\qquad s_i=\frac{2S_i}{1+cS_1S_2}\quad(i=1,2). π 0 ( c ) = 4 1 + c S 1 S 2 , s i = 1 + c S 1 S 2 2 S i ( i = 1 , 2 ) . Since the baseline factor cancels in score expectations, we have E π 0 ( c ) s i = 0 \mathbb{E}_{\pi_0^{(c)}}s_i=0 E π 0 ( c ) s i = 0 E π 0 ( c ) [ ( S j / 2 ) s i ] = 1 i = j \mathbb{E}_{\pi_0^{(c)}}[(S_j/2)s_i]=\mathbf{1}_{i=j} E π 0 ( c ) [( S j /2 ) s i ] = 1 i = j A A A Y Y Y 1 / 2 1/2 1/2 s 1 , s 2 s_1,s_2 s 1 , s 2 U = A U=A U = A V = Y V=Y V = Y s 1 s_1 s 1 U = V = A U=V=A U = V = A ℓ 1 , ℓ 2 \ell_1,\ell_2 ℓ 1 , ℓ 2 x x x π 0 ( c ) ( 1 + ℓ 1 s 1 + ℓ 2 s 2 ) \pi_0^{(c)}(1+\ell_1s_1+\ell_2s_2) π 0 ( c ) ( 1 + ℓ 1 s 1 + ℓ 2 s 2 )
P ( S 1 = σ 1 , S 2 = σ 2 ∣ X ) = 1 + c σ 1 σ 2 + 2 ℓ 1 σ 1 + 2 ℓ 2 σ 2 4 , w = 1 2 + ℓ 1 , m = 1 2 + ℓ 2 , P(S_1=\sigma_1,S_2=\sigma_2\mid X)=\frac{1+c\sigma_1\sigma_2+2\ell_1\sigma_1+2\ell_2\sigma_2}{4},\qquad w=\frac{1}{2}+\ell_1,\qquad m=\frac{1}{2}+\ell_2, P ( S 1 = σ 1 , S 2 = σ 2 ∣ X ) = 4 1 + c σ 1 σ 2 + 2 ℓ 1 σ 1 + 2 ℓ 2 σ 2 , w = 2 1 + ℓ 1 , m = 2 1 + ℓ 2 , N = c 4 − ∫ p ℓ 1 ℓ 2 , D = 1 4 − ∫ p ℓ 1 2 . \mathcal{N}=\frac{c}{4}-\int p\ell_1\ell_2,\qquad\mathcal{D}=\frac{1}{4}-\int p\ell_1^2. N = 4 c − ∫ p ℓ 1 ℓ 2 , D = 4 1 − ∫ p ℓ 1 2 . Lower bracket when θ < 1 / 2 \theta<1/2 θ < 1/2 If β ≤ α \beta\le\alpha β ≤ α ( α , β ) (\alpha,\beta) ( α , β ) c = 0 c=0 c = 0 ( ℓ 1 , ℓ 2 ) = ( u , v ) (\ell_1,\ell_2)=(u,v) ( ℓ 1 , ℓ 2 ) = ( u , v ) ( U , V , W , λ ) = ( A , Y , A Y , − 1 ) (U,V,W,\lambda)=(A,Y,AY,-1) ( U , V , W , λ ) = ( A , Y , A Y , − 1 ) β > α \beta>\alpha β > α ( α , α ) (\alpha,\alpha) ( α , α ) c = 1 / 4 c=1/4 c = 1/4 ( ℓ 1 , ℓ 2 ) = ( u + v , 0 ) (\ell_1,\ell_2)=(u+v,0) ( ℓ 1 , ℓ 2 ) = ( u + v , 0 ) ( U , V , W , λ ) = ( A , A , A , − 1 ) (U,V,W,\lambda)=(A,A,A,-1) ( U , V , W , λ ) = ( A , A , A , − 1 ) s 1 s_1 s 1 D = 1 \mathcal{D}=1 D = 1 [ r − , r + ] = [ p − , p + ] [r_-,r_+]=[p_-,p_+] [ r − , r + ] = [ p − , p + ] θ \theta θ P O W ( α , β ) \mathcal{P}_{\mathrm{OW}}(\alpha,\beta) P OW ( α , β ) p − ≤ p ≤ p + p_-\le p\le p_+ p − ≤ p ≤ p + w ∈ H α ( H 0 ) w\in\mathcal{H}^{\alpha}(H_0) w ∈ H α ( H 0 ) m ∈ H β ( H 0 ) m\in\mathcal{H}^{\beta}(H_0) m ∈ H β ( H 0 ) ϵ ≤ w ≤ 1 − ϵ \epsilon\le w\le1-\epsilon ϵ ≤ w ≤ 1 − ϵ ( α , β ) (\alpha,\beta) ( α , β ) k 0 = 1 / 2 < H 0 k_0=1/2<H_0 k 0 = 1/2 < H 0 γ K = α \gamma_K=\alpha γ K = α w w w γ K = β \gamma_K=\beta γ K = β m m m ( α , α ) (\alpha,\alpha) ( α , α ) w w w γ K = α \gamma_K=\alpha γ K = α m = 1 / 2 m=1/2 m = 1/2 ∥ m ∥ C β = 1 / 2 < H 0 \lVert m\rVert_{C^\beta}=1/2<H_0 ∥ m ∥ C β = 1/2 < H 0 Lemma 7.2 , the hard laws lie in P O W ( α , β ) \mathcal{P}_{\mathrm{OW}}(\alpha,\beta) P OW ( α , β ) n n n N \mathcal{N} N D \mathcal{D} D t n = c 1 a n ( θ , τ p ) t_n=c_1a_n(\theta,\tau_p) t n = c 1 a n ( θ , τ p )
In the first case, eventually D ≥ 1 / 8 \mathcal{D}\ge1/8 D ≥ 1/8
∣ N − γ O W 4 ∣ = ∣ γ O W ∣ ∫ p u 2 ≤ 8 ∥ u ∥ ∞ 3 ∥ v ∥ ∞ ≤ C n − θ − 2 α / d = o ( t n ) . \left|\mathcal{N}-\frac{\gamma_{\mathrm{OW}}}{4}\right|=|\gamma_{\mathrm{OW}}|\int pu^2\le8\lVert u\rVert_\infty^3\lVert v\rVert_\infty\le Cn^{-\theta-2\alpha/d}=o(t_n). N − 4 γ OW = ∣ γ OW ∣ ∫ p u 2 ≤ 8 ∥ u ∥ ∞ 3 ∥ v ∥ ∞ ≤ C n − θ − 2 α / d = o ( t n ) . The map γ ↦ γ / 4 \gamma\mapsto\gamma/4 γ ↦ γ /4 [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] γ O W \gamma_{\mathrm{OW}} γ OW N = 1 / 16 \mathcal{N}=1/16 N = 1/16 D ∈ [ 3 / 16 , 1 / 4 ] \mathcal{D}\in[3/16,1/4] D ∈ [ 3/16 , 1/4 ] γ O W ∈ [ 1 / 4 , 1 / 3 ] \gamma_{\mathrm{OW}}\in[1/4,1/3] γ OW ∈ [ 1/4 , 1/3 ] D = 1 / ( 16 γ O W ) \mathcal{D}=1/(16\gamma_{\mathrm{OW}}) D = 1/ ( 16 γ OW ) [ 1 / 4 , 1 / 3 ] [1/4,1/3] [ 1/4 , 1/3 ] Lemma A.3 (b), with k = 0 k=0 k = 0 T 1 T_1 T 1 T 0 = γ O W T_0=\gamma_{\mathrm{OW}} T 0 = γ OW γ O W \gamma_{\mathrm{OW}} γ OW t n / 4 t_n/4 t n /4 Lemma A.2 gives the lower bracket.
Root-n n n Let P t \mathcal{P}_t P t p = 1 p=1 p = 1 { 1 + t S 1 S 2 } / 4 \{1+tS_1S_2\}/4 { 1 + t S 1 S 2 } /4 ∣ t ∣ < 1 / 2 \lvert t\rvert<1/2 ∣ t ∣ < 1/2 w = m = 1 / 2 w=m=1/2 w = m = 1/2 N = t / 4 \mathcal{N}=t/4 N = t /4 D = 1 / 4 \mathcal{D}=1/4 D = 1/4 γ O W = t \gamma_{\mathrm{OW}}=t γ OW = t P O W ( α , β ) \mathcal{P}_{\mathrm{OW}}(\alpha,\beta) P OW ( α , β ) 1 + t S 1 S 2 ≥ 1 / 2 1+tS_1S_2\ge1/2 1 + t S 1 S 2 ≥ 1/2 Lemma 5.1 implies the root-n n n
References 