本文精选了统计学国际顶刊《Annals of Statistics》近期发表的论文，提供统计学研究领域最新的学术动态。

Orthogonal statistical learning

原刊和作者：

Annals of Statistics Volume 51 Issue 3

Dylan J. Foster （Microsoft Research）

Vasilis Syrgkanis （Stanford University）

Abstract

We provide nonasymptotic excess risk guarantees for statistical learning in a setting where the population risk with respect to which we evaluate the target parameter depends on an unknown nuisance parameter that must be estimated from data. We analyze a two-stage sample splitting meta-algorithm that takes as input arbitrary estimation algorithms for the target parameter and nuisance parameter. We show that if the population risk satisfies a condition called Neyman orthogonality, the impact of the nuisance estimation error on the excess risk bound achieved by the meta-algorithm is of second order. Our theorem is agnostic to the particular algorithms used for the target and nuisance and only makes an assumption on their individual performance. This enables the use of a plethora of existing results from machine learning to give new guarantees for learning with a nuisance component. Moreover, by focusing on excess risk rather than parameter estimation, we can provide rates under weaker assumptions than in previous works and accommodate settings in which the target parameter belongs to a complex nonparametric class. We provide conditions on the metric entropy of the nuisance and target classes such that oracle rates of the same order, as if we knew the nuisance parameter, are achieved.

Link: https://doi.org/10.1214/23-AOS2258

Optimal high-dimensional and nonparametric distributed testing under communication constraints

原刊和作者：

Annals of Statistics Volume 51 Issue 3

Botond Szabó （Bocconi University）

Lasse Vuursteen （Delft University of Technology）

Harry van Zanten （Vrije Universiteit Amsterdam）

Abstract

We derive minimax testing errors in a distributed framework where the data is split over multiple machines and their communication to a central machine is limited to b bits. We investigate both the d- and infinite-dimensional signal detection problem under Gaussian white noise. We also derive distributed testing algorithms reaching the theoretical lower bounds.

Our results show that distributed testing is subject to fundamentally different phenomena that are not observed in distributed estimation. Among our findings we show that testing protocols that have access to shared randomness can perform strictly better in some regimes than those that do not. We also observe that consistent nonparametric distributed testing is always possible, even with as little as one bit of communication, and the corresponding test outperforms the best local test using only the information available at a single local machine. Furthermore, we also derive adaptive nonparametric distributed testing strategies and the corresponding theoretical lower bounds.

Link: https://doi.org/10.1214/23-AOS2269

Optimal Permutation Estimation in CrowdSourcing problems

原刊和作者：

Annals of Statistics Volume 51 Issue 3

Emmanuel Pilliat （Université Montpellier）

Alexandra Carpentier （Universität Potsdam）

Nicolas Verzelen （Université Montpellier）

Abstract

Motivated by crowdsourcing applications, we consider a model where we have partial observations from a bivariate isotonic n×d matrix with an unknown permutation π∗ acting on its rows. Focusing on the twin problems of recovering the permutation π∗ and estimating the unknown matrix, we introduce a polynomial-time procedure achieving the minimax risk for these two problems, this for all possible values of n, d, and all possible sampling efforts. Along the way we establish that, in some regimes, recovering the unknown permutation π∗ is considerably simpler than estimating the matrix.

Link: https://doi.org/10.1214/23-AOS2271

Total positivity in multivariate extremes

原刊和作者：

Annals of Statistics Volume 51 Issue 3

Frank Röttger （Université de Genève）

Sebastian Engelke （Université de Genève）

Piotr Zwiernik （University of Toronto）

Abstract

Positive dependence is present in many real world data sets and has appealing stochastic properties that can be exploited in statistical modeling and in estimation. In particular, the notion of multivariate total positivity of order 2 (MTP2) is a convex constraint and acts as an implicit regularizer in the Gaussian case. We study positive dependence in multivariate extremes and introduce MTP2, an extremal version of EMTP2. This notion turns out to appear prominently in extremes, and in fact, it is satisfied by many classical models. For a Hüsler–Reiss distribution, the analogue of a Gaussian distribution in extremes, we show that it is EMTP2 if and only if its precision matrix is a Laplacian of a connected graph. We propose an estimator for the parameters of the Hüsler–Reiss distribution under EMTP2 as the solution of a convex optimization problem with Laplacian constraint. We prove that this estimator is consistent and typically yields a sparse model with possibly nondecomposable extremal graphical structure. Applying our methods to a data set of Danube River flows, we illustrate this regularization and the superior performance compared to existing methods.

Link: https://doi.org/10.1214/23-AOS2272

A power analysis for model-X knockoffs with ?p-regularized statisticss

Annals of Statistics Volume 51 Issue 3

Asaf Weinstein （Hebrew University of Jerusalem）

Weijie J. Su （University of Pennsylvania）

Ma?gorzata Bogdan （University of Wroclaw）

Rina Foygel Barber （University of Chicago）

Emmanuel J. Candès （Stanford University）

Abstract

Variable selection properties of procedures utilizing penalized-likelihood estimates is a central topic in the study of high-dimensional linear regression problems. Existing literature emphasizes the quality of ranking of the variables by such procedures as reflected in the receiver operating characteristic curve or in prediction performance. Specifically, recent works have harnessed modern theory of approximate message-passing (AMP) to obtain, in a particular setting, exact asymptotic predictions of the type I, type II error tradeoff for selection procedures that rely on ?p-regularized estimators.

In practice, effective ranking by itself is often not sufficient because some calibration for Type I error is required. In this work, we study theoretically the power of selection procedures that similarly rank the features by the size of an ?p-regularized estimator, but further use Model-X knockoffs to control the false discovery rate in the realistic situation where no prior information about the signal is available. In analyzing the power of the resulting procedure, we extend existing results in AMP theory to handle the pairing between original variables and their knockoffs. This is used to derive exact asymptotic predictions for power. We apply the general results to compare the power of the knockoffs versions of Lasso and thresholded-Lasso selection, and demonstrate that in the i.i.d. covariate setting under consideration, tuning by cross-validation on the augmented design matrix is nearly optimal. We further demonstrate how the techniques allow to analyze also the Type S error, and a corresponding notion of power, when selections are supplemented with a decision on the sign of the coefficient.

Link: https://doi.org/10.1214/23-AOS2274