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

Abstract : Let $(Y,(X_i)_{i\in\mathcal{I}})$ be a zero mean Gaussian vector and $V$ be a subset of $\mathcal{I}$. Suppose we are given $n$ i.i.d. replications of the vector $(Y,X)$. We propose a new test for testing that $Y$ is independent of $(X_i)_{i\in \mathcal{I}\backslash V}$ conditionally to $(X_i)_{i\in V}$ 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 neighbourhood of a Gaussian graphical model. The procedure is based on a model of Gaussian regression with random Gaussian covariates. We give non asymptotic 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. Besides, it allows us to derive non asymptotic minimax rates of testing in this setting. Finally, we carry out a simulation study in order to evaluate the performance of our procedure.
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https://hal.inria.fr/inria-00186919
Contributor : Nicolas Verzelen <>
Submitted on : Friday, May 23, 2008 - 12:14:24 PM
Last modification on : Wednesday, October 14, 2020 - 4:21:35 AM
Long-term archiving on: : Friday, September 24, 2010 - 10:38:07 AM

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  • HAL Id : inria-00186919, version 4
  • ARXIV : 0711.2119

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Nicolas Verzelen, Fanny Villers. Goodness-of-fit Tests for high-dimensional Gaussian linear models. [Research Report] RR-6354, INRA; INRIA. 2007, pp.46. ⟨inria-00186919v4⟩

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