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Goodness-of-fit tests for high-dimensional Gaussian linear models. (English) Zbl 1183.62074
Summary: Let \((Y, (X_i)_{1\leq i\leq p})\) be a real zero mean Gaussian vector and \(V\) be a subset of \(\{1, \dots , p\}\). Suppose we are given \(n\) i.i.d. replications of this vector. We propose a new test for testing that \(Y\) is independent of \((X_i)_{i\in \{1, \dots , p\}\setminus V}\) conditionally to \((X_i)_{i\in V}\) 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.

62G10 Nonparametric hypothesis testing
62J05 Linear regression; mixed models
62H20 Measures of association (correlation, canonical correlation, etc.)
05C90 Applications of graph theory
62H15 Hypothesis testing in multivariate analysis
Full Text: DOI
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