Finite Population Linear Prediction May Allow,

In this paper, admissibility of linear prediction in finite populations with arbitrary rank under matrix loss function (d-Qy)(d - Qy)' and quadratic loss function (d -Qy)'A(d -Qy). where A 0 is a known matrix, are investigated respectively.First, in this paper we consider the general Gauss-Markov linear model: y = Xβ + e , E(e) = 0, Var(e) = V ( or 2V), V 0. Under the above stated two loss functions, the definitions of admissible prediction are given, and the necessary and sufficient conditions for a linear prediction to be admissible in the class of homogeneous linear predictions and the class of nonhomogeneous linear predictions are obtained.Secondly, in the general random effects linear model : y=Xβ+ e , E(β',e')'=(a'A',0')', Cov(β',e')'=2(or 2), 0 and under the above stated loss functions, we give the definitions of admissible prediction and obtain the necessary and sufficient conditions for a linear prediction to be admissible in the class of homogeneous linear predictions , the class of nonhomogeneous linear predictions , and the class that contains all predictions. At the same time, we prove that the best linear unbiased prediction of a linear predictable variable Qy and the best linear unbiased prediction βn=(X'sV-1Xs)-1 X'sVs-1ys of regression coefficients of the finite populations βn= (X'V-1X)-1X'V-1y are admissible under matrix loss function and in the class of the linear predictions and the class that contains all predictions, and under quadratic loss function in the class of the linear predictions.