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

Nicolas Verzelen 1, 2 Fanny Villers 3
1 SELECT - Model selection in statistical learning
Inria Saclay - Ile de France, LMO - Laboratoire de Mathématiques d'Orsay, CNRS - Centre National de la Recherche Scientifique : UMR
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.
Type de document :
Rapport
[Research Report] RR-6354, INRA; INRIA. 2007, pp.46
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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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