The Annals of Statistics

Goodness-of-fit tests for high-dimensional Gaussian linear models

Nicolas Verzelen and Fanny Villers

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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.

Article information

Ann. Statist., Volume 38, Number 2 (2010), 704-752.

First available in Project Euclid: 19 February 2010

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Zentralblatt MATH identifier

Primary: 62J05: Linear regression
Secondary: 62G10: Hypothesis testing 62H20: Measures of association (correlation, canonical correlation, etc.)

Gaussian graphical models linear regression multiple testing ellipsoid adaptive testing minimax hypothesis testing minimax separation rate goodness-of-fit


Verzelen, Nicolas; Villers, Fanny. Goodness-of-fit tests for high-dimensional Gaussian linear models. Ann. Statist. 38 (2010), no. 2, 704--752. doi:10.1214/08-AOS629.

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