Relationship Of BLUPs For Linear Functions Under Mixed Linear Models

In this paper, we give a explicit representation of the best linear unbiased predictor (BLUP) for the linear function of fixed effects and random effects f(L, N) under a general mixed linear model. Using matrix rank method, we present the relationships of best linear unbiased predictors (BLUPs) for f(L, N) between two different general linear mixed models. Besides, we also consider the relationships of best linear unbiased predictors for f(L, N) between two different restricted mixed linear models.This paper is divided into 3 parts:In part 1, we consider the research background and present stituation of predictors in the linear mixed model. At the same time, introduce the linear mixed model and matrix basics needed. In part 2, we consider a general mixed linear model without any rank assumptions to the covariance matrix and without any restrictions on the correlation between the random effects vector and the random errors vector. We show a new necessary and sufficient condition for linear statistic Ay to be the BLUP for linear function of fixed effects and realized values of random effects f(L,N). Moreover, we derive some necessary and sufficient conditions for the BLUP(f(L, N)) under M1 continue to be the BLUP(f(L, N)) under the M2, as well as some necessary and sufficient conditions for the equivalence of BLUP(f(L,N)) under the two linear mixed models M1 and M2.In part 3, we mainly consider a restricted mixed linear model without any rank assumptions to the covariance matrix and without any restrictions on the correlation between the random effects vector and the random errors vector. We show some necessary and sufficient conditions for Ty+c to be the best linear unbiased predictor (BLUP) of linear function f(L, N) with fixed effects and realized values of random effects. Moreover, we derive some equalities of BLUP(f(L,N)) under different mixed linear models.