Covariance Matrix Estimate In Linear Mixed Models

The thesis is concerned with the further argumentation of the spectral decomposition estimation in linear mixed models. It is known that the spectral decomposition estimate has many properties.Thereinto, for the spectral decomposition estimate of the covariance matrix ,we can gain the risk functions under some losses.The thesis is based on the above-mentioned property.The discussions are as follows:Firstly,we propose new kind of estimates of covariance matrix based on its spectral decomposition estimate (its weighted type),and obtain their corresponding risk functions under Stein loss and Ertopy loss.Thereby we discuss the best estimate in these new estimates in virtue of the functions.Furthermore we gain the new estimation of the covariance (the weighted type of its SDE) and call the reduced es-timate.Under Etropy loss, this new estimate of the covariance matrix is its spectral decomposition estimate.But its new estimate is differ from the spectral decomposition estimate of the covariance matrix. At the same time,we prove the property that the risk function of the new estimate of the covariance matrix is less than its spectral decomposition estimate,ANOVAE and MINQUE under Stein loss in some models.Secondly,we give the reduced estimates of the latent roots and the variance components and prove some properties of these reduced estimates. In addition,it is proved that the reduced estimates of the latent roots and the variance components are superrior to their spectral decomposition estimates under the meaning of MSE.