Least Square Estimation of the Parameters of the Multivariate Normal Covariance Matrix

Keywords: Least square estimation, Cholesky decomposition, unconstrained parameterization

Abstract: Modeling the covariance matrix of a random vector has received much attention in longitudinal and multivariate data analysis. In general, the covariance matrix is non-diagonal, to model it we remove the positive definiteness constraint and provide an unconstrained parameterization of the covariance matrix, using Cholesky decomposition (Pourahmadi, 1999, Biometrika). This gives us a unique unit lower triangular matrix T and a diagonal matrix D with positive entries such that TST' = D. This new set of parameters are unconstrained and meaningful, thus they can be modeled using covariates. In this talk we focus only on linear models for these new parameters. In a simulation study we show that the least square method provides good estimates of these new parameters of the covariance matrix.