Selection of Loss Function in Covariance Structure Analysis: Case of the Spherical Model

Abstract

In this paper, we derive the asymptotic properties of estimators obtained from various kinds of loss functions in covariance structure analysis. We first show that the estimators except for OLS-based loss functions have the same asymptotic distribution when the dimension of the covariance matrix, p, is fixed and the sample size n tends to infinity. Then, focusing on the spherical model, we show that this equivalence does not hold when both n and p become larger. Specifically, we show that some estimators lose consistency, and even consistent estimators have different asymptotic variances. Among the estimators considered, the maximum likelihood estimator shows the best performance, while the less famous invGLS(ub) estimator performs better than the commonly used GLS estimator. We also demonstrate the validity of the likelihood ratio test for the spherical and diagonal models in a high-dimensional framework.</p