Note on asymptotic null distributions of LR statistics for testing covariance matrix under growth curve model when the number of the observation points is large

Abstract

This paper is concerned with the testing problem about the covariance matrix under growth curve model. The testing problems treated in this paper are the following problems: (i) the problem of testing that a covariance matrix is equal to a specified positive definite matrix, (ii) the problem of testing for sphericity of the covariance matrix, and (iii) the problem of intraclass model for the covariance matrix. We give asymptotic distributions of the null distributions for the likelihood ratio statistics under an asymptotic framework that the total sample size $n$ go to infinity, the number of the observation points p go to infinity, and $p/n$ go to a constant $c\in (0,1)$. Simulation reveals that the proposed approximations have good accuracies compared with the classical chi-square approximations.