GOODNESS-OF-FIT TESTS FOR HIGH-DIMENSIONAL GAUSSIAN LINEAR MODELS

• 成果类型: Article
• 署名作者: Verzelen, Nicolas; Villers, Fanny
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite Paris Saclay; Universite Paris Saclay; INRAE
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/08-AOS629
• 发表日期: 2010
• 页码: 704-752
• 关键词: VARIABLE SELECTION; graphs

摘要

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 anew test for testing that Y is independent of (X-i)(i is an element of{1,...,p}\V) conditionally to (X-i)(i is an element of 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.