Testing linear hypotheses in high-dimensional regressions

TL;DR

This paper develops a high-dimensional correction to Wilk's likelihood ratio test for multivariate linear models, enabling accurate hypothesis testing when both data dimension and sample size are large.

Contribution

The authors propose a novel asymptotic correction based on random matrix theory for Wilk's test in high-dimensional settings, improving its accuracy and applicability.

Findings

Corrected LRT is asymptotically Gaussian under the null hypothesis.

Simulation results show the corrected test maintains size and power for large p and n.

Standard chi-square approximation fails in high-dimensional contexts.

Abstract

For a multivariate linear model, Wilk's likelihood ratio test (LRT) constitutes one of the cornerstone tools. However, the computation of its quantiles under the null or the alternative requires complex analytic approximations and more importantly, these distributional approximations are feasible only for moderate dimension of the dependent variable, say $p\le 20$. On the other hand, assuming that the data dimension $p$ as well as the number $q$ of regression variables are fixed while the sample size $n$ grows, several asymptotic approximations are proposed in the literature for Wilk's $\bLa$ including the widely used chi-square approximation. In this paper, we consider necessary modifications to Wilk's test in a high-dimensional context, specifically assuming a high data dimension $p$ and a large sample size $n$. Based on recent random matrix theory, the correction we propose to Wilk's test is asymptotically Gaussian under the null and simulations demonstrate that the corrected LRT has very satisfactory size and power, surely in the large $p$ and large $n$ context, but also for moderately large data dimensions like $p=30$ or $p=50$. As a byproduct, we give a reason explaining why the standard chi-square approximation fails for high-dimensional data. We also introduce a new procedure for the classical multiple sample significance test in MANOVA which is valid for high-dimensional data.