Research On The Parameter Estimation Of The Mixed-Linear Models

Because of the wide usage of mixed-linear model in biology, medicine, computer and microwave engineering and other fields, the statistic research on this model received increasing attention. The unknown parameters of this model are divided into two categories: one is the fixed effects, and the other is the variance components. With regard to the estimation of the variance components, many statisticians proposed a number of approaches, such as the Analysis of Variance Estimator, Maximum Likelihood Estimator, Restricted MLE, and the Minimum Norm Quadratic Unbiased Estimator, etc. For these estimators, computation of the estimators of the fixed effects and the variance components are separated. Except the Analysis of Variance Estimator, these approaches all need to solve a non-linear equation, which does not have explicit solution, and only has an iteration solution in general.Recently, Wang Songgui and some other people put forward a new estimation approach for the fixed effects and variance components, which is called spectral decomposition estimator. The peculiarity of this new approach is that the estimation for the fixed effects is the linear estimation which possesses a good quality for estimation. The second chapter of the present thesis attempts to make a systematic introduction and generalization on its methods, qualities and applications and propose the future research direction. In addition, some hot issues concerning the current researches and the newest research results about the mixed-linear models are being appropriately discussed and explained.In chapter three, Bayes estimators of variance components are derived for mixed-linear models with two variance components under the weighted square loss function, and the empirical Bayes estimators are constructed by the kernel estimation method of multivariate density and its mixed partial derivatives, and the convergence rates of this estimator are established.Finally, as far as the general mixed-linear model is concerned, the present thesis explores the two-staged prediction of the linear combination about the fixed effects and the random effects and meanwhile an approximate computation formula is derived for the mean square error of the two-staged prediction which paves the way for comparing and contrasting the advantages and disadvantages of different variance components.