Optimal rates for independence testing via $U$-statistic permutation tests
type: Item
instance of: scholarly article
publication date: 2020-01-15T00:00:00Z
publication date: 2021-12-03T00:00:00Z
arXiv ID: 2001.05513
arXiv classification: math.ST
arXiv classification: stat.ME
arXiv classification: stat.ML
arXiv classification: stat.TH
Mathematics Subject Classification ID: 62C20
Mathematics Subject Classification ID: 62G10
Mathematics Subject Classification ID: 62H20
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 62G09
Mathematics Subject Classification ID: 62G20
author: Thomas B. Berrett
author: Ioannis Kontoyiannis
author: Richard J. Samworth
author: Richard Samworth
DOI: 10.1214/20-AOS2041
DOI: 10.48550/ARXIV.2001.05513
MaRDI profile type: Publication
title: Optimal rates for independence testing via \(U\)-statistic permutation tests
zbMATH Open document ID: 1504.62060
published in: The Annals of Statistics
full work available at URL: https://arxiv.org/abs/2001.05513
full work available at URL: 20-AOS2041.full
review text: The authors study the problem of independence testing in a general framework, where the data consists of independent copies of a pair \((X,Y)\) taking values in a separable measure space \((\mathcal{X},\mathcal{Y})\), equipped with a \(\sigma\)-finite measure \(\mu\). Assuming that the joint distribution of \((X,Y)\) has a density \(f\) with respect to \(\mu\), one may define a measure of dependence \(D(f)\), given by the squared \(L^2(\mu)\) distance between the joint density and the product of its marginal densities. This satisfies the natural requirement that \(D(f)=0\) if and only if \(X\) and \(Y\) are independent. However, Theorem 1 reveals that it is not possible to construct a valid independence test with nontrivial power against all alternatives satisfying a lower bound on \(D(f)\). This motivates to introduce classes satisfying an additional Sobolev-type smoothness condition as well as boundedness conditions on the joint and marginal densities. The first main goal of this work is determination of the minimax separation rate of independence testing over these classes, and to this end, a new permutation test of independence based on a \(U\)-statistic estimator of \(D(f)\) is defined. Further, Theorem 2 in Section 3 provides a very general upper bound on the separation rate of independence testing. Note that the framework is broad enough to include both discrete and absolutely continuous data, as well as data that may take values in infinite-dimensional spaces. The authors show how the bound can be simplified in many special cases, and, in Section 4, how to construct adaptive versions of their tests that incur only a small loss in effective sample size. Moreover, in Section 5, matching lower bounds in several instances is provided, allowing to conclude that suggested USP test attains the minimax optimal separation rate for independence testing in such settings. In Section 6, an approximation to the power function of the test at local alternatives is elucidate…
zbMATH DE Number: 7438256
zbMATH Keywords: independence testing
zbMATH Keywords: minimax separation rates
zbMATH Keywords: permutation tests
zbMATH Keywords: Stein's method
zbMATH Keywords: U-statistics
zbMATH Keywords: \(U\)-statistic permutation (USP) test
reviewed by: Jaromír Antoch
describes a project that uses: energy
describes a project that uses: copula
describes a project that uses: dHSIC
describes a project that uses: R
describes a project that uses: IndepTest
describes a project that uses: AS 159
describes a project that uses: GeneralisedCovarianceMeasure
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Optimal rates for independence testing via $U$-statistic permutation tests (OpenAlex ID)
OpenAlex ID: W3213451519