Non-binding Parameters. Mean Covariance Model Maximum Likelihood Estimate

How to estimate variance-covariance matrix is a problem that has received substantial consideration in recent years. Many of the theoretical and practical difficulties can be attributed to the fact that the co variance matrix is constrained to be non-negative definite; furthermore, when the data come from unbalanced repeated measures, the problem of estimating a covariance matrix becomes increasingly difficult. Recently several methods of covariance estimation have used an unconstrained parameterization. These methods have been facilitated by the matrix logarithmic transformation, spectral decomposition and Cholesky decomposition. Pourahmadi develops maximum likelihood estimation of linear models for multivariate normal data and he shows that when the observations come from a balanced design, the estimators are consistent and normally asymptotic. However, the methodology for the unbalanced data is undeveloped.In this article, we extend the work of Pourahmadi (1999,2000). Specifically, when the data come from i.n.i.d. populations, we use covariates to model the mean vector and covariance matrix. Under the assumption of normality, we use an unconstrained parameterization and maximum likelihood to estimate the covariance matrix. Furthermore, we show that our estimators are consistent and normally asymptotic, a fact which has not been investigated. Of course, similar to the case of balanced data, our method ensures that the estimated covariance matrix is positive definite.The outline of this article is as follows:Chapter 1 provides the background of covariance problem and several unconstrained parameterizations involved in this article; Chapter 2 deals with the work of Pourahmadi(1999,2000) and we improve his method to calculate some variables involved in the estimation process; Chapter 3 develops the maximum likelihood estimation for the general mean-covariance model and provides the numerical simulation with our method.