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

2007/11/14 by Nicolas Verzélen, Verzelen, Nicolas, Fanny Villers +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.0711.2119

openalex publication_date 2007/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (Y,(X_i)_i\inI) be a zero mean Gaussian vector and V be a subset of I. Suppose we are given n i.i.d. replications of the vector (Y,X). We propose a new test for testing that Y is independent of (X_i)_{i∈ I\backslash 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 neighbourhood of a Gaussian graphical model. The procedure is based on a model of Gaussian regression with random Gaussian covariates. We give non asymptotic 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. Besides, it allows us to derive non asymptotic minimax rates of testing in this setting. Finally, we carry out a simulation study in order to evaluate the performance of our procedure.

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