Testing composite hypotheses, Hermite polynomials and optimal estimation of a nonsmooth functional

Cai, T. Tony; Low, Mark G.

doi:10.1214/10-aos849

Cited by 65 publications

(124 citation statements)

References 12 publications

Supporting

Mentioning

119

Contrasting

Order By: Relevance

“…Proof. By [CL11], proof of Theorem 3, (see also [WY18], Lemma 14), if P and Q are supported on [−V, V ], then χ 2 (P * N (0, 1), Q * N (0, 1)) ≤ e V 2 /2 ∞ =1 ∆ 2 ! .…”

Section: Proof Let πmentioning

confidence: 99%

Uncoupled isotonic regression via minimum Wasserstein deconvolution

Rigollet

Weed

2019

Information and Inference: A Journal of the IMA

View full text Add to dashboard Cite

Isotonic regression is a standard problem in shape-constrained estimation where the goal is to estimate an unknown nondecreasing regression function f from independent pairs (While this problem is well understood both statistically and computationally, much less is known about its uncoupled counterpart where one is given only the unordered sets {x 1 , . . . , x n } and {y 1 , . . . , y n }. In this work, we leverage tools from optimal transport theory to derive minimax rates under weak moments conditions on y i and to give an efficient algorithm achieving optimal rates. Both upper and lower bounds employ moment-matching arguments that are also pertinent to learning mixtures of distributions and deconvolution.

show abstract

“…Proof. By [CL11], proof of Theorem 3, (see also [WY18], Lemma 14), if P and Q are supported on [−V, V ], then χ 2 (P * N (0, 1), Q * N (0, 1)) ≤ e V 2 /2 ∞ =1 ∆ 2 ! .…”

Section: Proof Let πmentioning

confidence: 99%

Uncoupled isotonic regression via minimum Wasserstein deconvolution

Rigollet

Weed

2019

Information and Inference: A Journal of the IMA

View full text Add to dashboard Cite

show abstract

“…Later, Lepski et al [76] considered estimating the L r , r ≥ 1 norm of a regression function, and utilized trigonometric approximation. Cai and Low [77] used best polynomial approximation to estimate the ℓ 1 norm of a Gaussian mean.…”

Section: Motivation Methodology and Related Workmentioning

confidence: 99%

“…The next two lemmas from Cai and Low [77] are simple facts we will utilize in the analysis of our estimators.…”

Section: Lemma 15mentioning

confidence: 99%

Minimax Estimation of Functionals of Discrete Distributions

Jiao

Venkat

Han

et al. 2015

IEEE Trans. Inform. Theory

144

View full text Add to dashboard Cite

Abstract-We propose a general methodology for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters, and elaborate on the case of discrete distributions, where the support size S is unknown and may be comparable with or even much larger than the number of observations n. We treat the respective regions where the functional is "nonsmooth" and "smooth" separately. In the "nonsmooth" regime, we apply an unbiased estimator for the best polynomial approximation of the functional whereas, in the "smooth" regime, we apply a biascorrected version of the Maximum Likelihood Estimator (MLE).We illustrate the merit of this approach by thoroughly analyzing the performance of the resulting schemes for estimating two important information measures: the entropyWe obtain the minimax L2 rates for estimating these functionals. In particular, we demonstrate that our estimator achieves the optimal sample complexity n S/ ln S for entropy estimation. We also demonstrate that the sample complexity for estimating Fα(P ), 0 < α < 1, is n S 1/α / ln S, which can be achieved by our estimator but not the MLE. For 1 < α < 3/2, we show the minimax L2 rate for estimating Fα(P ) is (n ln n) −2(α−1) for infinite support size, while the maximum L2 rate for the MLE is n −2(α−1) . For all the above cases, the behavior of the minimax rate-optimal estimators with n samples is essentially that of the MLE (plug-in rule) with n ln n samples, which we term "effective sample size enlargement".We highlight the practical advantages of our schemes for the estimation of entropy and mutual information. We compare our performance with various existing approaches, and demonstrate that our approach reduces running time and boosts the accuracy. Moreover, we show that the minimax rate-optimal mutual information estimator yielded by our framework leads to significant performance boosts over the Chow-Liu algorithm in learning graphical models. The wide use of information measure estimation suggests that the insights and estimators obtained in this work could be broadly applicable.

show abstract

“…Remark 2.6. The above lemmas concern testing two and multiple composite hypotheses about an arbitrary operator F : Θ → (R d , d) defined on some parameter space Θ, which generalize the classical Le Cam's method and the Fano's method (Yu, 1997;Tsybakov, 2009), as well as the ideas of Cai and Low (2011). The proofs of these lemmas can be found in the Supplementary Material (Ma et al, 2019b).…”

Section: Minimax Lower Bound Optimality and Adaptivitymentioning

confidence: 99%

Optimal Permutation Recovery in Permuted Monotone Matrix Model

Cai

2020

Journal of the American Statistical Association

View full text Add to dashboard Cite

Motivated by applications in metagenomics, we consider the permuted monotone matrix model Y = ΘΠ + Z, where Y ∈ R n×p is observed, Θ ∈ R n×p is an unknown signal matrix with monotone rows, Π ∈ R p×p is an unknown permutation matrix, and Z ∈ R n×p is the noise matrix. This paper studies the estimation of the extreme values associated to the signal matrix Θ, including its first and last columns, as well as their difference (the range vector). Treating these estimation problems as compound decision problems, minimax rate-optimal and adaptive estimators are constructed using spectral column sorting. Novel techniques that can be effective in estimating an arbitrary high-dimensional nonlinear operator are developed to establish minimax lower bounds, including generalized Le Cam's method and Fano's method. Numerical experiments using simulated and synthetic microbiome metagenomic data are presented, showing the superiority of the proposed methods over the alternatives. The methods are illustrated by comparing the growth rates of gut bacteria between inflammatory bowel disease patients and normal controls.

show abstract

Testing composite hypotheses, Hermite polynomials and optimal estimation of a nonsmooth functional

Cited by 65 publications

References 12 publications

Uncoupled isotonic regression via minimum Wasserstein deconvolution

Uncoupled isotonic regression via minimum Wasserstein deconvolution

Minimax Estimation of Functionals of Discrete Distributions

Optimal Permutation Recovery in Permuted Monotone Matrix Model

Contact Info

Product

Resources

About