Studies Of Some Issues Of The Parametric Empirical Bayes Estimation In Linear Model

This dissertation focus on the construction and property of Bayes estimation andparametric empirical Bayes (PEB) estimation of parameters in linear model.First, we introduce some basic concepts and current research results of Bayes anal-ysis and parametric empirical Bayes method, then we describe the definitions of supe-riority criterions of estimation and the paper's construction.In Chapter two, we derive the Bayes estimator and PEB estiamtors of the errorvariance in a normal linear regression model when the regression coefficient is seenas an unknown nuisaince parameters and the error variance has conjugate prior distri-bution. The superiority of the PEB estimators over the least squares(LS) estimator isinvestigatedunderthemeansquareerror(MSE)criterion, andtheasymptoticpropertiesare also studied. When the hyperparameters of prior distribution are all unknown, wereconstruct the PEB estimators and use simulation methods to obtain the superiority ofthe PEB estimators over LS estimator.In Chapter three, when the regression coefficients and error-variance have thenormal-invert Gamma prior distributions, the simultaneous Bayes estimators of the re-gression coefficients and error-variance are derived in linear model. The superioritiesof the Bayes estimator over the LS estimator of regression coefficients are investigatedin terms of the mean square error matrix (MSEM) criterion and Pitman Closeness (PC)criterion, and the superiority of the Bayes estimator of error-variance over LS estimatoris also discussed under the mean square error (MSE) criterion. When the hyperparam-eters are partly known, the simultaneous PEB estimators of the regression coefficientsand error-variance are constructed, and the superiority of PEB estimators over LS esti-mators of regression coefficients is investigated in terms of the MSEM criterion. Thesuperiority of PEB estimator over LS estimator of the error-variance is discussed underthe MSE criterion. Finally, when all hyperparameters are unknown, we reconstruct thePEB estimators of regression coefficients and error-variance and study the superiorityof them over LS estimator by simulation methods.In Chapter four, in one-way classification random effects model of variance com-ponents, we derive the Bayes estimators of variance components when they have conju-gatepriordistributions. ThesuperiorityoftheBayesestimatorsofvariancecomponentsover Analysis of Variance (ANOVA) estimators under MSE criterion are obtained interms of MSE criterion. When the hyperparameters are partly unknown, we construct the PEB estimators of variance components and study the superiorities of PEB esti-mators over ANOVA estimators under MSE criterion. We also obtain the asymptoticproperties of PEB estimators. When all hyperparameters are unknown, the PEB estima-tors of variance components are reconstructed and the simulation results of them overANOVA estimators are gotten.Finally, we summarize our research work, point out the significance and feature ofour work, and offer some problems which we will study in future.