Type: Article
Publication Date: 2013-09-26
Citations: 32
DOI: https://doi.org/10.1080/02331888.2012.708031
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 hypothesis requires complex analytic approximations, and more importantly, these distributional approximations are feasible only for moderate dimension of the dependent variable, say p≤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 Λ 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 hypothesis 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 such as 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 multivariate analysis of variance which is valid for high-dimensional data. Keywords: high-dimensional datamultivariate regressionmultivariate analysis of varianceWilk's testmultiple sample significance testrandom matrices AMS 2000 Subject Classifications : Primary 62H15secondary 62H10 Acknowledgements The authors acknowledge the support from the following research grants: NSFC grant 11171057 (Z.D. Bai), NSFC grant 11101181 and RFDP grant 20110061120005 (D. Jiang), HKU Start-up Fund (J. Yao) and NSFC-11171058 and NECT-11-0616 (S. Zheng).