| Linear regression model is one of the most important models in modern statistics. Estimating regression coefficient is the most fundamental issue in linear regression model from theory and application. According to Guass-Markov theory, we know that least squares estimation is the best estimator and has wide applications.However, with the development of admissibility theory and the study in large regression problem which have more variables, people find that least squares estimation become very bad in some cases. So people begin to seek new estimation to replace least squares estimation. Starting from reducing the mean squared error of least squares estimation of regression coefficientβ, People present many important estimations, such as ridge estimation, Stein estimation, Liu estimation and so on. Among these estimation, Stein estimation is the most simple and the first estimation in biased estimation, it occupies an important position in the developmental history of biased estimation.In chapter three, starting from the minimum generalized mean square error of a class of linear estimator, we propose a new Stein estimation (β|∧)~*( k) .In MSE criterion, the sufficient condition that (β|∧)~*( k) is more superior than least squares estimation is got. Then we introduce one criterion named Pitman criterion which is used to evaluate superiority of estimation. After analysis of rationality of Pitman criterion, we also get the sufficient condition that (β|∧)~*( k) is more superior than least squares estimation.Bayesian methods have already penetrated into all fields of statistics and become one of important component of statistics. For different prior distribution, we get the Bayes estimation of linear regression coefficient and prove that it is uniformly more superior than least squares estimation in Pitman criterion. Whenσ2 is known, Stein estimation is Bayesian, which show that the new estimation (β|∧)~*( k) that we have proposed is also Bayes estimation at some prior distribution. |
