Study On The Restricted Biased Estimation In The Linear Model

The research into the biased estimation in the linear model is all the time one of the most popular issues of regression analysis in statistics. While dealing with the multicollinearity of design matrix X , the ordinary least squares estimation is always helpless. The linear biased estimation is the most direct method in ameliorating the ordinary least squares estimation. As the gradual maturity of the biased estimation of the linear regression model without additional linear equation restrictions, a great deal of statistical problems such as in the experiment design, hypothesis test, the models of variance analysis and the covariance analysis and so on, which show in practice the investigative significances and the applied values of the restricted linear regression, need to be regressed with several additional linear restrictions. Like the ordinary least squares estimation, the ordinary restricted least squares estimation applied widely is also disadvantageous for dealing with the multicollinearity of design matrix. As a result, a great many researchers recently try to find out a better method to improve the ordinary restricted least squares estimation.In this dissertation, we try to find out some biased estimations better than the ordinary restricted least squares estimation. Besides, the estimations proposed in the dissertation will be introduced into the hypothesis tests. In a word, three tasks will be done in the dissertation:(1) Introducing the global restrictions ( or elliptical restrictions ), we obtain a restricted ridge estimation by minimizing the sum of squared residuals with restricted conditions. After that, We analyse theoretically the character of the new estimation in its bias, stabilization and superiority compared with the ordinary restricted least squares estimation. What's more, we conclude a sufficient and necessary condition for selecting the parameter k in which the restricted ridge estimation proposed is surely superior to the ordinary restricted least squares estimation. And we give an illustration.(2) Introducing the additional stochastic linear restriction, we obtain a restricted unified biased estimation by minimizing the sum of squared residuals with restricted conditions and deduce some other estimations. Then we also analyse theoretically the character of the new estimation in its bias, stabilization and superiority compared with the ordinary restricted least squares estimation. Further more, we deduce the elliptical range in which the restricted unified biased estimation proposed is surely superior to the...