%0 Journal Article
%T On the Stochastic Restricted Class Estimator and Stochastic Restricted Class Estimator in Linear Regression Model
%A Jibo Wu
%J Journal of Applied Mathematics
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/173836
%X The stochastic restricted class estimator and stochastic restricted class estimator are proposed for the vector of parameters in a multiple linear regression model with stochastic linear restrictions. The mean squared error matrix of the proposed estimators is derived and compared, and some properties of the proposed estimators are also discussed. Finally, a numerical example is given to show some of the theoretical results. 1. Introduction The problem of multicollinearity or the ill-conditioned design matrix in linear regression model is very well known in statistics. In order to overcome this problem, different remedies have been introduced. One of the most important estimation methods is to consider biased estimators, such as the principal component regression (PCR) estimator [1], the ridge estimator (ORE) by Hoerl and Kennard [2], the class estimator [3], the Liu estimator (LE) by Liu [4], the class estimator [5], the class estimator [6], and the principal component Liu-type estimator [7]. An alternative method to deal with multicollinearity problem is to consider parameter estimation with some restrictions on the unknown parameters, which may be exact or stochastic restrictions [8]. When stochastic additional restrictions on the parameter vector are supposed to hold, Durbin [9], Theil and Goldberger [10], and Theil [11] proposed the ordinary mixed estimator (OME). By grafting the ordinary regression ridge estimator and LE into the mixed estimation, Li and Yang [12] and Hubert and Wijekoon [13] introduced a stochastic restricted ridge estimator (SRRE) and stochastic restricted Liu estimator (SRLE), respectively, and Liu et al. [14] proposed the weighted mixed almost unbiased ridge estimator in linear regression model. In this paper, in order to overcome multicollinearity, we introduce a stochastic restricted class estimator and a stochastic restricted class estimator for the vector of parameters in a linear regression model when additional stochastic linear restrictions are assumed to hold. Performance of the proposed estimators with respect to the mean squared error matrix (MSEM) criterion is discussed. The rest of the paper is organized as follows. The model specifications and the new estimators are introduced in Section 2. Then, the superiority of the proposed estimators is discussed in Section 3 and a numerical example is given to illustrate the behavior of the estimators in Section 4. Finally, some conclusion remarks are given in Section 5. 2. Model Specifications and the Estimators Consider the linear regression model where is an vector of
%U http://www.hindawi.com/journals/jam/2014/173836/