Testing linear hypotheses in high-dimensional regressions

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.

Accepted 02/2012 for publication in "Statistics". 20 pages, 2 pages and 2 tables

Cited by in corpus (10)

• On the sphericity test with large-dimensional observations
• Asymptotic Linear Spectral Statistics for Spiked Hermitian Random Matrix Models
• Likelihood Ratio Test in Multivariate Linear Regression: from Low to High Dimension
• CLT for linear spectral statistics of large dimensional sample covariance matrices with dependent data
• Hypergeometric Functions of Matrix Arguments and Linear Statistics of Multi-Spiked Hermitian Matrix Models
• High-dimensional general linear hypothesis tests via non-linear spectral shrinkage
• Moderate-Dimensional Inferences on Quadratic Functionals in Ordinary Least Squares
• Linear Eigenvalue Statistics of $X{X}^{\prime }$ matrices
• Modified Pillai's trace statistics for two high-dimensional sample covariance matrices
• On the Phase Transition of Wilk's Phenomenon