Research On Biased Estimators Of Parameters In Linear Model

Linear model is one of the important models in modern statistics, which has many applications in biology, medicine, economy, management, geologic, weather, agriculture, industry and so on, and it is a wide application research branch in modern statistics. The problem of parameter estimation is the most part in the area of the linear model. In this dissertation, we mainly study the biased estimation of parameter in linear model, and obtain some new biased estimators and result.In the first part, we consider the parameter estimation in linear model. Firstly, a new two-parameter estimator is proposed by combining the ridge estimator and the Liu estimator. This new estimator is a general estimator which includes the ordinary least squares estimator, the ridge estimator and the Liu estimator as special case. Necessary and sufficient conditions for the superiority of the new estimator over ordinary least squares estimaor, the ridge estimator and the Liu estimator in the mean squared error matrix sense are derived. We also obtain the estimators of the biasing parameters. Secondly, we introduce principal component two-parameter estimator, by combining the principal component regression estimator and the two-parameter estimator. The superiority of the new estimator over the principal component regression estimator, the r ? k class estimator, the r ? d class estimator and the two-parameter estimator are discussed with respect to the mean squared error matrix criterion. Finally, we propose a new estimator by jackknifing the modified ridge estimator. We show that our new estimator is superior to the modified ridge estimator and the ordinary least squares estimator in the mean squared error matrix and mean squared error sense. Furthermore, for the new proposed estimators, numerical example and Monte Carlo simulation study are given to illustrate our findings.In the second part, we consider the linear model with stochastic linear restrictions. When the prior information and the sample information are not equally important in practice, we introduce the following new estimators, namely weighted mixed Liu estimator and weighted mixed ridge estimator. Necessary and sufficient conditions for the superiority of the new estimator over the weighted mixed estimator in the mean squared error matrix sense are derived when the parameter restrictions are correct and are not correct. Numerical example and Monte Carlo simulation are given to illustrate some of the theoretical results. When the suspicion of stochastic constraints occurring, we consider the preliminary test approach to the estimation of the regression parameter in linear model with multivariate Student-t distribution. The preliminary test estimators based on the Wald (W), Likelihood Ratio (LR) and Lagrangian Multiplier (LM) tests are given under the suspicion of stochastic constraints occurring. The bias, mean square error matrix and weighted mean square error of the proposed estimators are derived and compared. The conditions of superiority of the proposed estimators are obtained. Finally, we conclude that the optimum choice of the level of significance becomes the traditional choice by using the W test.In the third part, we consider the linear model with linear equality restrictions. We introduce the preliminary test approach to the estimation of the regression parameter in linear model with normal error and multivariate Student-t error, and propose the preliminary test two-parameter estimator based on F test. The superiority of the new estimator over the two-parameter estimator, the restricted two-parameter estimator, the preliminary test ridge estimator and the preliminary test Liu estimator are discussed with respect to the mean squared error criterion under the null and alternative hypotheses. Moreover, the preliminary test two-parameter estimators based on the W, LR and LM tests are given, when it is suspected that the regression parameter may be restricted to a subspace. The bias and the mean square error of the proposed estimators are derived and compared. The conditions of superiority of the proposed estimators are obtained. Finally, we conclude that the optimum choice of the level of significance becomes the traditional choice by using the W test.