Asymptotic distribution for the proportional covariance model

TL;DR

This paper derives the asymptotic distribution of maximum likelihood estimators in the proportional covariance model under multivariate normality, using Cholesky parametrization and the delta method, with an application to covariance homogeneity testing.

Contribution

It introduces a novel derivation of the asymptotic distribution for the proportional covariance model using Cholesky parametrization and the delta method.

Findings

Asymptotic distribution of MLEs for proportional covariance model derived.

Distribution for Cholesky root of covariance matrix obtained.

Application to testing covariance matrix homogeneity.

Abstract

Asymptotic distribution for the proportional covariance model under multivariate normal distributions is derived. To this end, the parametrization of the common covariance matrix by its Cholesky root is adopted. The derivations are made in three steps. First, the asymptotic distribution of the maximum likelihood estimators of the proportionality coefficients and the Cholesky inverse root of the common covariance matrix is derived by finding the information matrix and its inverse. Next, the asymptotic distributions for the case of the Cholesky root of the common covariance matrix and finally for the case of the common covariance matrix itself are derived using the multivariate $\delta$-method. As an application of the asymptotic distribution derived here, a hypothesis for homogeneity of covariance matrices is considered.