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

Nicolas N. Verzelen (1, 2) , Fanny Villers (3)

1 MISTEA - Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie
2 SELECT - Model selection in statistical learning
3 MIA-JOUY - Unité de biométrie et intelligence artificielle de Jouy

Let (Y,(Xi )1≤i≤p) 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 )i∈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 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.

Mots clés: LINEAR REGRESSION; MULTIPLE TESTING; ELLIPSOID; ADAPTIVE TESTING; MINIMAX HYPOTHESIS TESTING; GOODNESS-OF-FIT

Mots clés: GAUSSIAN GRAPHICAL MODELS; MINIMAX SEPARATION RATE; MODELE GRAPHIQUE GAUSSIEN; VALIDITE DE L’AJUSTEMENT

Domaines: Mathématiques [math]; Informatique [cs]

Nicolas N. Verzelen, Fanny Villers. Goodness-of-fit tests for high-dimensional Gaussian linear models. Annals of Statistics, 2010, 38 (2), pp.704-752. ⟨10.1214/08-AOS629⟩. ⟨hal-02661047⟩