A Linear Regression Model Unbiased Estimator

Starting with the equivalence condition of admissibility, this paper discusses respectively the linear biased estimate of the regression coefficient of thee models: the G-M Linear Regression Model, the Multivariate Linear Regression and the Mixed-effect Coefficient Linear Model.In the G-M Linear Regression M.odel(y,XJ3,a-2In cr2 > 0), this paper discusses the estimate class of the regression coefficient ft ,such as = AXy (A is a matrix of p).ln fact, this estimate class is an essentially complete subclass of all the linear estimate class of ft. The necessary and sufficient condition of the statement, = AX'y is a linear admissible estimate, is that A is a symmetric matrix and that the relative characteristic values of A related to(X'X) all lie in [0,1]. When the error distribution is the Normal Distribution, the admissible estimate class such as fi = AX'y is equivalent to the class of the Bayes Estimate and the Limited Bayes Estimate. When (.4) = k, the principal component e-stimate fi'k = PPXy of has the properties of r-minmax and .4,0, -maxmin in the admissible estimate class such as = AX'y ,which generalizes some findings in paper [3].In the Multivariate Linear Model (Y, XB, Z <8> In), this paper discusses the estimate class of parameter matrix B such as B = AXYC(A is a nonzero symmetric matrix of p , C = (C,,C2,-,C)B) is a nonzero matrix of m), and the estimate class of parameter vector such as = AXrCi (i = 1,2,-,m). When Y (XB,cr2Z0 /?| a2 > 0 is unknown ) The necessary and sufficient condition of the statement, B = AXYC is a linear admissible estimate of B, is (A, A are the relative characteristic values of C'Z related to S and A related to (X'X}^ respectively). The paper [2] puts forward an estimate = (XXyXYC, of , defined by (3.3)) and an estimate B = (X'X)~] XYC of B . It also puts forward the pre-test hypotheses HiQ,Hi0 and the test functions Oy(.y), O( O) according to Hi0, Hit). This paper proves that the noncentral chi-square distribution class and the noncentral F-distribution class both are the Monotone Likelihood Ratio Distribution Classes. So that O>0,. (y) are the UMP test of Hi0, Hi0. It also proves that, is not related to y .0 and is a linear admissible estimate of , but B is not a linear admissible estimate of B . Based on those, this paper gives two kinds of improved estimates 5* and B(k). Respectively, it gives their advantages and disadvantages. On the term of Mean Square Error, B' is firmly and uniformly better than B .In the Mixed-effect Coefficient Linear Model, two kinds of General-ized Stein Estimates d(K) = QKiQ'di and d(G) = QGQ'd are put forw-ard according to two kinds of the LS estimates of parameter d .It offers two kinds of values of K and G by minimizing the mean square error and minimizing the unbiased estimate of mean square error.