Admissibility Of Linear Predictor In Multivariate Linear Model With Linear Equality Constraint

Linear model is an important branches of mathematical statistics, which be of distant history, a great deal of theory and some applied value. Of late thirty years, H. Bolfarine etc have attached importance to optimal prediction problem in linear model, and obtained best linear unbiased predictors, minimax predictors, Bayes predictors and simple project predictors of population total and finite population regression coefficients. Specifically, with the increasing popularity of giant computer, the research in this domain is forging rapidly ahead. China's scholar study mainly parameter estimation in linear model instead of prediction. Yu Sheng-hua etc extend a part of H. Bolfarine's research, and translate the concept and theory of admissibility on parameter estimation into prediction in linear model. In papers, the unknown parameter in linear model is usually unrestrained, but sometimes we need study the linear model with restrained condition. For example, fixed effect and cross effect in variance analysis model and covariance analysis model always satisfy specified restrained condition. For this, we consider the following multivariate linear model with linear equality constraint and arbitrary rank:where Y is n×q random matrix of observations,σ₂>0 is an unknown parameter, B is a p×q unknown parameter matrix,εis an n×q random error matrix, X is an n×p known design matrix,Δ≥0 andΣ≥0 are known q×q and n×n matrices respectively, U is a s×p known constant matrix, and B∈ψ={B:UB =0}.We might as well assumeΔ≠0 andΣ≠0.In the second and third chapter of this paper, we investigate the above model under quadratic loss function and matrix loss function respectively. Definitions of conditional linear predictable variable and admissible linear predictor are given, and several necessary and sufficient conditions for a linear predictor of the conditional linear predictable variable to be admissible in the class of homogeneous linear predictors and the class of nonhomogeneous linear predictors are established respectively. In the fourth chapter we extend results in the third chapter to general growth curve model with linear equality constraint.