Adaptive tests of linear hypotheses by model selection

Abstract

We propose a new test, based on model selection methods, for testing that the expectation of a Gaussian vector with n independent components belongs to a linear subspace of ℝ n against a nonparametric alternative. The testing procedure is available when the variance of the observations is unknown and does not depend on any prior information on the alternative. The properties of the test are nonasymptotic and we prove that the test is rate optimal [up to a possible log(n) factor] over various classes of alternatives simultaneously. We also provide a simulation study in order to evaluate the procedure when the purpose is to test goodness-of-fit in a regression model.