TY - JOUR

T1 - Likelihood ratio tests for covariance matrices of high-dimensional normal distributions

AU - Jiang, Dandan

AU - Jiang, Tiefeng

AU - Yang, Fan

N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2012/8

Y1 - 2012/8

N2 - For a random sample of size n obtained from a p-variate normal population, the likelihood ratio test (LRT) for the covariance matrix equal to a given matrix is considered. By using the Selberg integral, we prove that the LRT statistic converges to a normal distribution under the assumption p/n→y∈(0, 1]. The result for y=1 is much different from the case for y∈(0, 1). Another test is studied: given two sets of random observations of sample size n 1 and n 2 from two p-variate normal distributions, we study the LRT for testing the two normal distributions having equal covariance matrices. It is shown through a corollary of the Selberg integral that the LRT statistic has an asymptotic normal distribution under the assumption p/n 1→y 1∈(0, 1] and p/n 2→y 2∈(0, 1]. The case for max{y 1, y 2}=1 is much different from the case max{y 1, y 2}<1.

AB - For a random sample of size n obtained from a p-variate normal population, the likelihood ratio test (LRT) for the covariance matrix equal to a given matrix is considered. By using the Selberg integral, we prove that the LRT statistic converges to a normal distribution under the assumption p/n→y∈(0, 1]. The result for y=1 is much different from the case for y∈(0, 1). Another test is studied: given two sets of random observations of sample size n 1 and n 2 from two p-variate normal distributions, we study the LRT for testing the two normal distributions having equal covariance matrices. It is shown through a corollary of the Selberg integral that the LRT statistic has an asymptotic normal distribution under the assumption p/n 1→y 1∈(0, 1] and p/n 2→y 2∈(0, 1]. The case for max{y 1, y 2}=1 is much different from the case max{y 1, y 2}<1.

KW - Gamma function

KW - High-dimensional data

KW - Selberg integral

KW - Testing on covariance matrices

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U2 - 10.1016/j.jspi.2012.02.057

DO - 10.1016/j.jspi.2012.02.057

M3 - Article

AN - SCOPUS:84859561923

VL - 142

SP - 2241

EP - 2256

JO - Journal of Statistical Planning and Inference

JF - Journal of Statistical Planning and Inference

SN - 0378-3758

IS - 8

ER -