Further Researches On Problems Of Estimation And Prediction In General Linear Models

In the linear model y = Xβ+ e, e～ (0,σ~2V), one task of the author in the present paper is to investigate the quadratic sufficiency with respect to the problem of nonnegative estimation for β'Hβ+ hσ~2 on the basis of observations y, and another is to predict the linear form Qy of y, and the quadratic form of e, the stochastic part of y.Firstly, the problem of nonnegative estimation for β'Hβ+ hσ~2 is discussed in the original model and its transformed model. The notion of quadratic sufficiency is considered in the sense of generality with respect to nonnegative unbiased estimation set and nonnegative biased estimation family, respectively, and the corresponding necessary and sufficient conditions for the transformation to be quadratically sufficient are given. Further, some special results are obtained. Moreover, a simulated example is given at the end of the chapter.Secondly, the problem of linear prediction for Qy in the original model and its transformed model is considered, the notion of linear sufficiency and the problem of comparison of linear models with respect to prediction problems are proposed and the corresponding necessary and sufficient conditions for the transformation to be linearly sufficient and for one model to be superior over another are given, respectively. Moreover, some special results of important cases are obtained.Thirdly, the problem of quadratic prediction for e'He in the original model is considered, a concept of invariant quadratic prediction is considered, a notion of optimal invariant quadratic unbiased predictor and another of optimal invariant quadratic biased predictor are proposed, and their descriptions are investigated. Moreover, some special results of noticeable cases are obtained, in which the minimum norm quadratic unbiased estimator of σ~2 under one sample being an optimal predictor for that of σ~2 under another sample is noticeable.