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

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.