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April 2010 Goodness-of-fit tests for high-dimensional Gaussian linear models
Nicolas Verzelen, Fanny Villers
Ann. Statist. 38(2): 704-752 (April 2010). DOI: 10.1214/08-AOS629


Let (Y, (Xi)1≤ip) be a real zero mean Gaussian vector and V be a subset of {1, …, p}. Suppose we are given n i.i.d. replications of this vector. We propose a new test for testing that Y is independent of (Xi)i∈{1, …, p}∖V conditionally to (Xi)iV against the general alternative that it is not. This procedure does not depend on any prior information on the covariance of X or the variance of Y and applies in a high-dimensional setting. It straightforwardly extends to test the neighborhood of a Gaussian graphical model. The procedure is based on a model of Gaussian regression with random Gaussian covariates. We give nonasymptotic properties of the test and we prove that it is rate optimal [up to a possible log(n) factor] over various classes of alternatives under some additional assumptions. Moreover, it allows us to derive nonasymptotic minimax rates of testing in this random design setting. Finally, we carry out a simulation study in order to evaluate the performance of our procedure.


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Nicolas Verzelen. Fanny Villers. "Goodness-of-fit tests for high-dimensional Gaussian linear models." Ann. Statist. 38 (2) 704 - 752, April 2010.


Published: April 2010
First available in Project Euclid: 19 February 2010

zbMATH: 1183.62074
MathSciNet: MR2604699
Digital Object Identifier: 10.1214/08-AOS629

Primary: 62J05
Secondary: 62G10 , 62H20

Keywords: Adaptive testing , ellipsoid , Gaussian graphical models , Goodness-of-fit , Linear regression , minimax hypothesis testing , minimax separation rate , multiple testing

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.38 • No. 2 • April 2010
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