Estimation Of Variance Components Matrix In Multivariate Linear Mixed Model

The aim of this thesis is to study the estimation of variance components matrices in general multivariate linear mixed model with two variance components matrices.The first chapter introduces the general multivariate linear mixed model with two variance components matrices, theories and methods which including least square method, the spectral decomposition estimation method and Wishart distribution definitions and properties.For the general multivariate linear mixed model with two variance components matrices, based on the spectral decomposition of the covariance matrix, the second chapter gives a new spectral decomposition estimation of variance components matrices.In the third chapter, the new spectral decomposition estimation and the analysis of variance estimation are compared, a relationship between the two estimates is established, and some conditions are found under which one estimate is superior to the other in terms of the squared error loss function. Through studying a class of estimators, in the sense of squared error risk, a better estimation is derived, which has smaller non-negative probability.In the fourth chapter, in view of the new spectral decomposition and analysis estimation of variance components matrices, we obtained the estimation of covariance matrix. In the sense of squared error risk, we discussed the superiority of SDE and ANOVAE when the model is balanced and unbalanced.