Further Study On The United Biased Estimation In The Linear Model

We know that the ordinary least squares estimation has good performance in the linear regression model, it's very important in estimation theory and practical application of linear statistical models. But when the design matrix X has the problem of multicollinearity, the ordinary least squares estimation shows apparently disadvantage, the linear biased estimation is the most direct method in ameliorating the ordinary least squares estimation. Then, Statisticians proposed many kinds of linear biased estimation, like Ridge Estimation, Principal Components Estimation, Liu Estimation and so on, and the United Biased Estimation includes the biased estimation which we common used. Due to its advantage in including so many biased estimations and can avoid repeated study, the United Biased Estimation has been studied widely. In this dissertation, we fatherly study the United Biased Estimation and these tasks will be done:1) Given to the important practical and application background of growth-curve model, and the problem of multicollinearity which lies in this model, we study the united biased estimation of growth-curve model, and propose the Multivariate United Biased Estimation, then we discuss it's admissibility, and deduce the elliptical range in which the Multivariate United Biased Estimation proposed is surely superior to the ordinary least squares estimation. Likewise, we often meet the restricted growth-curve model in practical application of linear model, when the multicollinearity happens in this model, we propose a new estimation---- Restricted Multivariate United Biased Estimation. Also, we discuss its admissibility and deduce the elliptical range in which the Restricted Multivariate United Biased Estimation proposed is surely superior to the restricted ordinary least squares estimation.2) There exist many literatures concerning the research that the biased estimators with certain restricted conditions outperform the least squares estimator under the criterion of the mean-square error. However, it's much harder to do the similar research under the pitman criterion. In this paper, we study the performances of both the UBE and the restricted UBE under the Pitman criterion, and then give the sufficient conditions under which the various biased estimators outperform the traditional LSE under the Pitman criterion.3) Based on hypothesis tests of the united biased estimation obtained in document,...