A Study On The Almost Unbiased Estimator In Linear Regression Model

The least squared estimator of parameter is not considered as a good estimator any more because of the existence of the Stein phenomena. How to improve the least squared estimator has becoming a hot topic.It was well known that the ill condition of design matrix led to the stein phenomena. To overcome this problem, many authors proposed some biased estimators to alternate the least squared estimator. In some sense, these estimators improved the least squared estimator.However, the biased estimator was apart from the true value of parameter, so some statisticians proposed the notation of the almost unbiased estimator. Though the almost unbiased estimator was still a biased estimator, it has smaller bias than corresponding biased estimator.In this paper, we first study the properties of the almost unbiased generalized ridge (AUGR) estimator and the almost unbiased generalized Liu (AUGL) estimator. The AUGR estimator and the AUGL estimator, the AUGR estimator and OLS estimator, the AUGL estimator and the OLS estimator are compared in terms of their mean square error matrix. Some sufficient and necessary conditions for AUGR estimator being better than the OLS estimator, the AUGL estimator being better than the OLS estimator, the AUGR estimator being better than the AUGL estimator are given, respectively. In addition, a necessary condition for AURR estimator being better than the OLS estimator is given. And then some ellipses that AUGR estimator is better than the OLS estimator and AUGL estimator is better than the OLS estimator are given, respectively.Second, the definition of the almost unbiased unified biased (AUUB) estimator is proposed. This definition includes the familiar almost unbiased estimators in literatures, and it is the unified expression of the familiar almost unbiased estimators. Followed the biased and variance are compared of AUUB estimator and the unified biased (UB) estimator, respectively. AUUB estimator has smaller bias than UB estimator and the variance of AUUB estimator is between the variance of UB estimator and 4 times of the variance of UB estimator. Finally the properties of AUUB estimator are discussed. The conclusion is gained that there are parameters made AUUB estimator is better than OLS estimator in terms of their mean square error. The sufficient and necessary condition that AUUB estimator is admissible is given. The ellipse is given that AUUB estimator is...