### Abstract

Let Np(μ,∑) be a p-dimensional normal distribution. Testing ∑ equal to a given matrix or (μ,∑) equal to a given pair through the likelihood ratio test (LRT) is a classical problem in the multivariate analysis. When the population dimension p is fixed, it is known that the LRT statistics go to 2-distributions. When p is large, simulation shows that the approximations are far from accurate. For the two LRT statistics, in the high-dimensional cases, we obtain their central limit theorems under a big class of alternative hypotheses. In particular, the alternative hypotheses are not local ones. We do not need the assumption that p and n are proportional to each other. The condition n - 1 > p ∞ suffices in our results.

Language | English (US) |
---|---|

Article number | 1750016 |

Journal | Random Matrices: Theory and Application |

Volume | 7 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2018 |

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### Keywords

- alternative hypothesis
- central limit theorem
- high-dimensional data
- Likelihood ratio test
- multivariate Gamma function
- multivariate normal distribution

### Cite this

*Random Matrices: Theory and Application*,

*7*(1), [1750016]. DOI: 10.1142/S2010326317500162

**A study of two high-dimensional likelihood ratio tests under alternative hypotheses.** / Chen, Huijun; Jiang, Tiefeng.

Research output: Contribution to journal › Article

*Random Matrices: Theory and Application*, vol 7, no. 1, 1750016. DOI: 10.1142/S2010326317500162

}

TY - JOUR

T1 - A study of two high-dimensional likelihood ratio tests under alternative hypotheses

AU - Chen,Huijun

AU - Jiang,Tiefeng

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Let Np(μ,∑) be a p-dimensional normal distribution. Testing ∑ equal to a given matrix or (μ,∑) equal to a given pair through the likelihood ratio test (LRT) is a classical problem in the multivariate analysis. When the population dimension p is fixed, it is known that the LRT statistics go to 2-distributions. When p is large, simulation shows that the approximations are far from accurate. For the two LRT statistics, in the high-dimensional cases, we obtain their central limit theorems under a big class of alternative hypotheses. In particular, the alternative hypotheses are not local ones. We do not need the assumption that p and n are proportional to each other. The condition n - 1 > p ∞ suffices in our results.

AB - Let Np(μ,∑) be a p-dimensional normal distribution. Testing ∑ equal to a given matrix or (μ,∑) equal to a given pair through the likelihood ratio test (LRT) is a classical problem in the multivariate analysis. When the population dimension p is fixed, it is known that the LRT statistics go to 2-distributions. When p is large, simulation shows that the approximations are far from accurate. For the two LRT statistics, in the high-dimensional cases, we obtain their central limit theorems under a big class of alternative hypotheses. In particular, the alternative hypotheses are not local ones. We do not need the assumption that p and n are proportional to each other. The condition n - 1 > p ∞ suffices in our results.

KW - alternative hypothesis

KW - central limit theorem

KW - high-dimensional data

KW - Likelihood ratio test

KW - multivariate Gamma function

KW - multivariate normal distribution

UR - http://www.scopus.com/inward/record.url?scp=85038372369&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85038372369&partnerID=8YFLogxK

U2 - 10.1142/S2010326317500162

DO - 10.1142/S2010326317500162

M3 - Article

VL - 7

JO - Random Matrices: Theory and Application

T2 - Random Matrices: Theory and Application

JF - Random Matrices: Theory and Application

SN - 2010-3263

IS - 1

M1 - 1750016

ER -