Testing linear hypotheses in high-dimensional regressions |
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Authors: | Zhidong Bai Dandan Jiang Jian-feng Yao |
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Institution: | 1. KLASMOE and School of Mathematics and Statistics , Northeast Normal University , 5268 People's Road, 130024 , Changchun , People's Republic of China;2. Institute of Mathematics, Jilin University , 2699 Qianjin Street, 130021 , Changchun , People's Republic of China;3. Department of Statistics and Actuarial Science , The University of Hong Kong , Pokfulam , Hong Kong |
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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. |
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Keywords: | high-dimensional data multivariate regression multivariate analysis of variance Wilk's test multiple sample significance test random matrices |
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