The Superiority Of Two Kinds Of Biased Estimators

In the regression coefficients estimator theories of linear regression model, the biased estimator is an important method to improve LSE . In this thesis two generalized kinds of biased estimators were put forward. One is a kind of linear biased estimator (k) = [I + k(X'X)-1 D]-1, where const k > 0 and D is p x p regular matrix, and the other is a kind of nonlinear biased estimator , where const k1 > 0, k2 > 0, B, D are p x p regular matrices and B is exchangeable with X'X. Then the superiority of them over the LSE was studied under MSE, GMSE and PC criterion.In the second chapter a kind of generalized linear biased estimator (k) and a kind of nonlinear biased estimator (k1,k2) were brought forward based on the smallest MSE. In the third chapter the superiority of (k) over the LSE was examined based on the GMSE. And then the finite-sample properties of (k) were discussed when D is exchangeable with X'X. At the end some applications of (k) were introduced in the common biased estimators. In the last chapter the asymptotic expansions of nonlinear biased estimator and generalized MSE were given out, and when error disturbances are sufficiently small , the necessary and sufficient condition of the superiority of (k1,k2) over the LSE is discussed, and then the superiority of (k1,k2) over the LSE was compared under GMSE and Pitman criterion.Because the superiority of (k1,k2) over the LSE is discussed in the whole parameter space, we know that LSE is not permissible in the set of nonlinear estimators. Hence I believe that the class of nonlinear biased estimator has great theoretic prospect and applied value.