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

Verzelen, Nicolas ; Villers, Fanny

Ann. Statist., Tome 38 (2010) no. 1, p. 704-752 / Harvested from Project Euclid

Résumé

Let (Y, (X_i)_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 (X_i)_{i∈{1, …, p}∖V} conditionally to (X_i)_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.

Détail

Publié le : 2010-04-15
Classification:  Gaussian graphical models,  linear regression,  multiple testing,  ellipsoid,  adaptive testing,  minimax hypothesis testing,  minimax separation rate,  goodness-of-fit,  62J05,  62G10,  62H20