Relationship of BLUP under general linear mixed model and its sub-model

In this study, a general linear mixed model and its sub-model obtained by deleting some fixed and random effects from this model are considered without making any restrictions on correlation of random effects and any full rank assumptions. The statistical results obtained from these models are not necessarily the same because of the structure of the models. Predictors of unknown parameter vectors have different expressions and different properties under the considered models. However, these predictors may have similar properties under some conditiones and assumptions. Therefore, examining the relationships between the prediction of linear combinations of joint unknown fixed and random vectors under these models is one of the main issues of statistical inference from these models. The main purpose of this study is to charaterize the relationships between the best linear unbiased predictors/estimators (BLUPs/BLUEs) of joint unknown parameter vectors under these two linear mixed models and to investigate the equality of the BLUPs/BLUEs under considered models. In order to obtain the results, some properties related to rank of matrices are used. There are five chapters in the study. In the first chapter, a literature review has been given on the considered models. In the second chapter, some basic definitions and theorems used throughout the study are given. Matrix equations of BLUP/BLUE for a general function of all unknown parameters and their properties are considerd in the third chapter. In the fourth chapter, the relationships between the BLUP/BLUE of the vectors of joint unknown parameters under these two linear mixed models are characterized and the equality of the BLUPs/BLUEs under the general linear mixed model and its sub-model was investigated. Then the result are given for some special cases. In the last chapter consists of conclusion and proposals. Keywords: BLUE, BLUP, general linear mixed model, random effects, rank, fixed effects