The Matrix-Logarithmic Covariance Model

Abstract

A flexible method is introduced to model the structure of a covariance matrix C and study the dependence of the covariances on explanatory variables by observing that for any real symmetric matrix A , the matrix exponential transformation, C = exp ( A ), is a positive definite matrix. Because there is no constraint on the possible values of the upper triangular elements on A , any possible structure of interest can be imposed on them. The method presented here is not intended to replace the existing special models available for a covariance matrix, but rather to provide a broad range of further structures that supplements existing methodology. Maximum likelihood estimation procedures are used to estimate the parameters, and the large-sample asymptotic properties are obtained. A simulation study and two real-life examples are given to illustrate the method introduced.