More Research On The Liu Estimator And Two-Parameter Estimator In The Linear Model

The research for parameter estimation in the linear model has been a hot issue instatistics, many scholars have made a lot of effort. In linear regression model, theproblem of multicollinearity (or ill-conditioned design matrix) and its statisticalconsequences are very well-known. The classic least squares method is inadequate indealing with the ill-conditioned design matrix, so many remedial measures have beenproposed to combat the problem. There are two methods which are more commonlyused, one is unbiased estimation, and the other is to consider parameter estimation withsome exact or stochastic restrictions on the unknown parameter. The linear biasedestimation is one of the most direct methods to overcome the ill-condition of designmatrix. In the area of biased estimators, some results are widely used, such asJames-Stein Shrinkage Estimator (JSSE), Principal Components Estimator (PCE),Ridge Estimator (RE), Liu Estimator (LE) and so on. Considering the restrictedcondition about parameter is one of most effective methods to solve the problem ofmulticollinearity. Just as the restricted least squares estimator (RLSE) and the mixedestimator (ME) improved the ordinary least squares estimator (OLSE) to a certainextent. In this paper, we consider the linear model under the conditions of theunrestricted and random restricted, and make a deep discussion on the basis of thepremier research. In fact, two tasks will be in the dissertation:For the general linear model, the two-parameter estimator (TPE) proposed by HuYang and Xinfeng Chang (2010) is discussed, which contains the common estimatorssuch as the ordinary linear square estimator, the ridge estimator and the Liu estimator. Inthis paper, necessary and sufficient conditions for the superiority of the two-parameterestimator over the ordinary linear square estimator, the ridge estimator, the Liuestimator and other two-parameter estimator under mean squared error matrix (MSEM)criterion is given by a different methods of proof. Furthermore, a numerical example isgiven to show the theoretical results.For the linear model with random restricted, a new stochastic weighted mixed Liuestimator is proposed, and the superiority of the new estimator is proved under meansquared error matrix criterion for the two cases in which the parametric restrictions aretrue or not, and the optimal choice of the parameter is given. Furthermore, we introducea new stochastic restricted two-parameter estimator which is a generalization of the mixed estimator and the two-parameter estimator. And we compare some relativeestimators under the mean squared error matrix criterion. What’s more, the numericalexample is given to illustrate the theoretical results respectively.