### 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 |

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### Keywords

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

### Cite this

**Limiting distributions of likelihood ratio test for independence of components for high-dimensional normal vectors.** / Qi, Yongcheng; Wang, Fang; Zhang, Lin.

Research output: Contribution to journal › Article

*Annals of the Institute of Statistical Mathematics*, vol. 71, no. 4, pp. 911-946. https://doi.org/10.1007/s10463-018-0666-9

}

TY - JOUR

T1 - Limiting distributions of likelihood ratio test for independence of components for high-dimensional normal vectors

AU - Qi, Yongcheng

AU - Wang, Fang

AU - Zhang, Lin

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.

KW - Central limit theorem

KW - Chi-square approximation

KW - Covariance matrix

KW - High-dimensional normal vector

KW - Independence

KW - Likelihood ratio test

UR - http://www.scopus.com/inward/record.url?scp=85046891128&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85046891128&partnerID=8YFLogxK

U2 - 10.1007/s10463-018-0666-9

DO - 10.1007/s10463-018-0666-9

M3 - Article

VL - 71

SP - 911

EP - 946

JO - Annals of the Institute of Statistical Mathematics

JF - Annals of the Institute of Statistical Mathematics

SN - 0020-3157

IS - 4

ER -