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2013 Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression
Laëtitia Comminges, Arnak S. Dalalyan
Electron. J. Statist. 7: 146-190 (2013). DOI: 10.1214/13-EJS766

Abstract

We consider the problem of testing a particular type of composite null hypothesis under a nonparametric multivariate regression model. For a given quadratic functional $Q$, the null hypothesis states that the regression function $f$ satisfies the constraint $Q[f]=0$, while the alternative corresponds to the functions for which $Q[f]$ is bounded away from zero. On the one hand, we provide minimax rates of testing and the exact separation constants, along with a sharp-optimal testing procedure, for diagonal and nonnegative quadratic functionals. We consider smoothness classes of ellipsoidal form and check that our conditions are fulfilled in the particular case of ellipsoids corresponding to anisotropic Sobolev classes. In this case, we present a closed form of the minimax rate and the separation constant. On the other hand, minimax rates for quadratic functionals which are neither positive nor negative makes appear two different regimes: “regular” and “irregular”. In the “regular" case, the minimax rate is equal to $n^{-1/4}$ while in the “irregular” case, the rate depends on the smoothness class and is slower than in the “regular” case. We apply this to the problem of testing the equality of Sobolev norms of two functions observed in noisy environments.

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Laëtitia Comminges. Arnak S. Dalalyan. "Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression." Electron. J. Statist. 7 146 - 190, 2013. https://doi.org/10.1214/13-EJS766

Information

Published: 2013
First available in Project Euclid: 24 January 2013

zbMATH: 1337.62090
MathSciNet: MR3020417
Digital Object Identifier: 10.1214/13-EJS766

Subjects:
Primary: 62G08, 62G10
Secondary: 62G20

Rights: Copyright © 2013 The Institute of Mathematical Statistics and the Bernoulli Society

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