On Some Aspects Of Statistical Inference In Linear Mixed Models And Multivariate Distribution

The thesis is concerned with the problems of statistic inference in some linear mixed models and multivariate distributions.For general linear mixed model, this thesis applies covariance improvement method given by Rao and obtains covariance improvement estimators of random effect variance components. Under mean square error(MSE) criterion, covariance improvement estimators dominate ANOVA estimators. The improvement estimator obtained by Mathew is a special case of them. In linear mixed model with two variance components, in order to obtain nonnegative estimators of variance components, nonnegative pre-test estimators of variance components are obtained and are better than ANOVA estimators (ANOVAE) under MSE criterion. At present, for the linear mixed model with two variance components, literatures only discuss the necessary and sufficient conditions for the spectral decomposition esti-mators(SDE) are equivalent to ANOVAE. In this thesis, we show that SDE have uniformly smaller mean square errors than ANOVAE, moreover, we extend these results to generalized spectral decomposition estimators(GSDE). Thereby, these facts indicate that the SDE not only have better statistic properties but also have uniformly smaller mean square errors than ANOVAE in an available group of linear mixed models. The other significative following work is to generalize the idea of SDE to multivariate linear mixed model and obtain the SDE of variance component matrices which uniformly dominate the ANOVAE either. Finally, we study the test problem of variance component in linear mixed model. Based on GSDE, an exact test of variance component is obtained by applying generalized p-value method. Simulation shows that the exact test dominates the type I error very well and has better power.For Panel data model, this thesis studies the comparisons about two stage esti-mator(TSE), least square estimator(LSE), Within estimator(WE) and Between es-timator(BE) of regression parameter under generalized mean square error(GMSE) criterion and Pitman criterion. Specially, it is significative that LSE uniformly dominates BE under Pitman criterion. Under Pitman criterion, a sufficient condition under which LSE dominates WE is obtained. Finally, we obtain two sufficient conditions under which TSE uniformly dominates BE and WE under GMSE criterion respectively. These conditions do not relate to the unknown parameters but the dimension of model, so it is easy to apply in practice. Next, we consider the relative efficiency of LSE against the beast linear unbiased estimator (BLUE). Applying the new results on upper bound of Norm-Type generalized Kantorovich inequality, the new results on the relative efficiency of LSE are obtained for Panel data model.For seemingly unrelated regression equations, some improved estimators of regression parameters are considered. By applying covariance adjusted method, we obtain a series of improvements on the estimators of regression parameters appeared in prevenient literatures and study the superiorities of these improved estimators. Specially, for two seemingly unrelated regression equations, we break normal constitutions and obtain new methods to construct covariance adjusted estimators. We divide sample into two groups and then obtain two independent unbiased estimators, when we construct the estimators of covariance matrix. Based on this process, we can reduce the order of moment and easily obtain better small sample properties while we study the superiorities of high order covariance adjusted estimators. By the way, we firstly study the superiorities of two steps covariance improved estimator under Pitman criterion and obtain a sufficient condition under which two steps covariance improved estimator dominates LSE.This thesis also considers some inference problems about parameters in some multivariate distributions. (1) For noncentrality parameter matrix in noncentral wishart distribution, based on usual unbiased estimator, a series of pre-test estimators are obtained. Under two usual quadratic loss functions, we obtain some simple conditions under which these pre-test estimators are uniformly better than unbiased estimator. As these conditions only depend on undetermined constants in pre-test estimators, it is easy to implement. (2) The isotonic estimations of mean vectors in two multivariate normal populations are investigated. From two cases that covariance matrices are known or unknown, we generalize the isotonic estimators in unitary case obtained by Oono and Shinozaki(2005) to multivariate case. When covariance matrices are known, we obtain isotonic estimators which uniformly dominate unrestricted maximum likelihood estimators. When covari- ance matrices are unknown and are restricted by partial order, we show that the isotonic estimators which uniformly dominate unrestricted maximum likelihood estimators. (3) Based on the nonlinear shrinkage idea raised by James and Stein, many authors extended this breakthrough to the improved estimation of multivariate normal mean and obtained many scalar shrinkage estimators and matricial shrinkage estimators. It is the first show that the modified Stein estimator is inadmissible when the scalar shrinkage is close to zero. Based on the modified Stein estimator, we introduce two adaptive scalar shrinkage coefficients into it and obtain two adaptive modified Stein estimators. The two adaptive modified Stein estimators are uniformly better than the Stein estimator and the modified Stein estimator relative to quadratic loss respectively.