Limiting distributions of likelihood ratio test for independence of components for high-dimensional normal vectors

Yongcheng Qi; Fang Wang; Lin Zhang

doi:10.1007/s10463-018-0666-9

Limiting distributions of likelihood ratio test for independence of components for high-dimensional normal vectors

Yongcheng Qi, Fang Wang, Lin Zhang

Mathematics & Statistics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

Consider a p-variate normal random vector. We are interested in the limiting distributions of likelihood ratio test (LRT) statistics for testing the independence of its grouped components based on a random sample of size n. In classical multivariate analysis, the dimension p is fixed or relatively small, and the limiting distribution of the LRT is a chi-square distribution. When p goes to infinity, the chi-square approximation to the classical LRT statistic may be invalid. In this paper, we prove that the LRT statistic converges to a normal distribution under quite general conditions when p goes to infinity. We propose an adjusted test statistic which has a chi-square limit in general. Our comparison study indicates that the adjusted test statistic outperforms among the three approximations in terms of sizes. We also report some numerical results to compare the performance of our approaches and other methods in the literature.

Original language	English (US)
Pages (from-to)	911-946
Number of pages	36
Journal	Annals of the Institute of Statistical Mathematics
Volume	71
Issue number	4
DOIs	https://doi.org/10.1007/s10463-018-0666-9
State	Published - Aug 7 2019

Bibliographical note

Funding Information:
The authors would like to thank three anonymous reviewers for their constructive comments and suggestions that have led to improvements in the paper. Qi’s research was supported by NSF Grant DMS-1005345, and Wang’s research was supported by NSFC Grant No. 11671021, NSFC Grant No. 11471222 and Foundation of Beijing Education Bureau Grant No. 201510028002.

Funding Information:
The authors would like to thank three anonymous reviewers for their constructive comments and suggestions that have led to improvements in the paper. Qi?s research was supported by NSF Grant DMS-1005345, and Wang?s research was supported by NSFC Grant No. 11671021, NSFC Grant No. 11471222 and Foundation of Beijing Education Bureau Grant No. 201510028002.

Publisher Copyright:
© 2018, The Institute of Statistical Mathematics, Tokyo.

Keywords

Central limit theorem
Chi-square approximation
Covariance matrix
High-dimensional normal vector
Independence
Likelihood ratio test

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10.1007/s10463-018-0666-9

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title = "Limiting distributions of likelihood ratio test for independence of components for high-dimensional normal vectors",

abstract = "Consider a p-variate normal random vector. We are interested in the limiting distributions of likelihood ratio test (LRT) statistics for testing the independence of its grouped components based on a random sample of size n. In classical multivariate analysis, the dimension p is fixed or relatively small, and the limiting distribution of the LRT is a chi-square distribution. When p goes to infinity, the chi-square approximation to the classical LRT statistic may be invalid. In this paper, we prove that the LRT statistic converges to a normal distribution under quite general conditions when p goes to infinity. We propose an adjusted test statistic which has a chi-square limit in general. Our comparison study indicates that the adjusted test statistic outperforms among the three approximations in terms of sizes. We also report some numerical results to compare the performance of our approaches and other methods in the literature.",

keywords = "Central limit theorem, Chi-square approximation, Covariance matrix, High-dimensional normal vector, Independence, Likelihood ratio test",

author = "Yongcheng Qi and Fang Wang and Lin Zhang",

note = "Funding Information: The authors would like to thank three anonymous reviewers for their constructive comments and suggestions that have led to improvements in the paper. Qi{\textquoteright}s research was supported by NSF Grant DMS-1005345, and Wang{\textquoteright}s research was supported by NSFC Grant No. 11671021, NSFC Grant No. 11471222 and Foundation of Beijing Education Bureau Grant No. 201510028002. Funding Information: The authors would like to thank three anonymous reviewers for their constructive comments and suggestions that have led to improvements in the paper. Qi?s research was supported by NSF Grant DMS-1005345, and Wang?s research was supported by NSFC Grant No. 11671021, NSFC Grant No. 11471222 and Foundation of Beijing Education Bureau Grant No. 201510028002. Publisher Copyright: {\textcopyright} 2018, The Institute of Statistical Mathematics, Tokyo.",

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AU - Qi, Yongcheng

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AU - Zhang, Lin

N1 - Funding Information: The authors would like to thank three anonymous reviewers for their constructive comments and suggestions that have led to improvements in the paper. Qi’s research was supported by NSF Grant DMS-1005345, and Wang’s research was supported by NSFC Grant No. 11671021, NSFC Grant No. 11471222 and Foundation of Beijing Education Bureau Grant No. 201510028002. Funding Information: The authors would like to thank three anonymous reviewers for their constructive comments and suggestions that have led to improvements in the paper. Qi?s research was supported by NSF Grant DMS-1005345, and Wang?s research was supported by NSFC Grant No. 11671021, NSFC Grant No. 11471222 and Foundation of Beijing Education Bureau Grant No. 201510028002. Publisher Copyright: © 2018, The Institute of Statistical Mathematics, Tokyo.

PY - 2019/8/7

Y1 - 2019/8/7

N2 - Consider a p-variate normal random vector. We are interested in the limiting distributions of likelihood ratio test (LRT) statistics for testing the independence of its grouped components based on a random sample of size n. In classical multivariate analysis, the dimension p is fixed or relatively small, and the limiting distribution of the LRT is a chi-square distribution. When p goes to infinity, the chi-square approximation to the classical LRT statistic may be invalid. In this paper, we prove that the LRT statistic converges to a normal distribution under quite general conditions when p goes to infinity. We propose an adjusted test statistic which has a chi-square limit in general. Our comparison study indicates that the adjusted test statistic outperforms among the three approximations in terms of sizes. We also report some numerical results to compare the performance of our approaches and other methods in the literature.

AB - Consider a p-variate normal random vector. We are interested in the limiting distributions of likelihood ratio test (LRT) statistics for testing the independence of its grouped components based on a random sample of size n. In classical multivariate analysis, the dimension p is fixed or relatively small, and the limiting distribution of the LRT is a chi-square distribution. When p goes to infinity, the chi-square approximation to the classical LRT statistic may be invalid. In this paper, we prove that the LRT statistic converges to a normal distribution under quite general conditions when p goes to infinity. We propose an adjusted test statistic which has a chi-square limit in general. Our comparison study indicates that the adjusted test statistic outperforms among the three approximations in terms of sizes. We also report some numerical results to compare the performance of our approaches and other methods in the literature.

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Limiting distributions of likelihood ratio test for independence of components for high-dimensional normal vectors

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