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

TL;DR

This paper introduces a new high-dimensional Gaussian independence test that is non-parametric, rate-optimal, and applicable to Gaussian graphical models, with proven theoretical properties and simulation validation.

Contribution

It develops a novel independence testing procedure for high-dimensional Gaussian vectors that does not rely on covariance assumptions and extends to Gaussian graphical models.

Findings

Test is rate optimal up to a log factor.

Procedure is non-asymptotic and minimax optimal.

Simulation results demonstrate effective performance.

Abstract

Let $(Y,(X_i)_{i\in\mathcal{I}})$ be a zero mean Gaussian vector and $V$ be a subset of $\mathcal{I}$. 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 $(X_i)_{i\in \mathcal{I}\backslash V}$ 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 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.