## 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) |
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Pages (from-to) | 911-946 |

Number of pages | 36 |

Journal | Annals of the Institute of Statistical Mathematics |

Volume | 71 |

Issue number | 4 |

DOIs | |

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