Study On Some Problems Of Biased Estimation

The linear model is one of the most commonly used model in statistics, which includes the linear regression model, analysis of variance model, analysis of covariance model and the variance components model, ect..In practical problems, especially in the solution of large regression problems with more independent variables, the design matrix inevitably exist multi-collinearity, when using the Least Squares Estimation to estimate the regression coefficient, which may make the sign of regression coefficient estimates contrary to the actual significance of the problem or absolute value of the estimated regression is very large.In order to solve multi-collinearity problem, statisticians have put forward biased estimate. Biased estimate sacrificed unbiasedmess in order to reduce the variance. In the sense of the mean-square error, the Biased Estimate can improve the Least Squares Estimation.In previous results of the statistical scholars, the paper makes further study on biased estimate and elaborates three solutions for the multi-collinearity problems which are Stein shrinkage estimation, Principal estimator and Ridge estimate. In the paper, it is introduced the three methods, summarized their natures. The paper improves Ridge Estimate which is the most frequently used in biased estimate and proves the better properties than the least squares estimate in the sense of the mean-square error.The paper promotes the hypothesis which the design matrix is full rank, heteroscedasticity and serial correlation to the assumption which are non-full rank, heteroscedasticity and serial correlation using the singular value decomposition theorem and the Schur decomposition theorem. Define Ridge Estimate on Moore-Penrose Inverse Matrix of General linear regression model and Ridge Estimation on M-P Inverse Matrix of Generalized Linear Regression Model discussed and proved that the nature of these two estimates. Both estimates are biased estimates, and the linear transformation of the least squares estimate, and compression estimation, and better than least squares estimation in the sense of the mean square error.