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

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.
Keywords :
Type de document :
Rapport
[Research Report] RR-6354, INRA; INRIA. 2007, pp.46
Domaine :
Liste complète des métadonnées

Littérature citée [29 références]

https://hal.inria.fr/inria-00186919
Contributeur : Nicolas Verzelen <>
Soumis le : vendredi 23 mai 2008 - 12:14:24
Dernière modification le : jeudi 11 janvier 2018 - 06:22:14
Document(s) archivé(s) le : vendredi 24 septembre 2010 - 10:38:07

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RR-testv3.pdf
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• HAL Id : inria-00186919, version 4
• ARXIV : 0711.2119

### Citation

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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