The Conditional Ridge-type Estimation In Singular Linear Model With Linear Equality Restrictions

Due to the disadvantage of the least squares estimate for dealing with the multicollinearity, the investigation into biased estimators in linear model is always one of the most popular issues in statistics. While the growing maturity of biased estimators in linear model without additional restrictions, various practical problems must be regressed with additional linear equality restrictions, which shows investigative significances and applied values. Like the least squares estimate, the restricted least squares estimate is also not ideal for dealing with the multicollinearity. As a result, a great many of researchers try to find out a better method to improve the restricted least squares estimate recently. The statisticians thus face the problem of choosing between the least squares estimate and a biased estimator. They also face the problem of choosing between two biased estimators. Therefore, the comparison among biased estimators has certain signification in theory and practice aspect.In this dissertation, considering the multicollinearity, we study the biased estimators in linear regression model much further. Four tasks will be done in the dissertation:①Considering the singular linear model with linear equality restrictions from the aspect of the biased estimators, we obtain a new restricted biased estimator: the conditional ridge-type estimation by minimizing the sum of squared residuals with restricted conditions. It has the property of avoiding the absolute bias of parameter vector due to the multicollinearity, and has the similar form with unrestricted biased estimators. Then we theoretically analyze the character of the new estimation in its bias, stabilization and monotone. In addition, we compare it with the ordinary restricted least squares estimate, and show its sufficient conditions under which it's superior over the restricted least squares estimate in terms of mean squares error and mean squares error matrix, respectively.②We study two pairs of biased estimators. We compare James—Stein estimator with ridge type estimator in terms of general mean squares error, and obtain the sufficient condition for the superiority of the James—Stein estimator over the ridge type estimator. Then we analyze the restricted ridge estimator and the restricted Liu estimator, and establish separate sufficient conditions for the superiority of the restricted ridge estimator over the restricted Liu estimator and the superiority of the restricted Liu estimator over the restricted ridge estimator in terms of mean squares error matrix. Furthermore, we obtain the parameter values of k and d in order to make use of the restricted Liu estimator and the restricted ridge estimator in practice.③Then, we give a numerical example. We compute the values of restricted ridge estimator and restricted Liu estimator under different values of k and d, compare their MSE, and prove these established sufficient conditions for the superiority of the restricted ridge estimator over the restricted Liu estimator and the superiority of the restricted Liu estimator over the restricted ridge estimator under mean squared error matrix.④Finally, we study the sample properties of the almost unbiased generalized Liu estimator, derive the exact general expressions for the moments of the almost unbiased generalized Liu estimator for individual regression coefficients, and export the exact expressions for the first two moments of the estimator.