Study On Biased Estimators In Linear Restricted Model

Linear model is an important branch in modern statistics and is widely used instatistics. In this paper, considering the multicollinearity, we mainly study the restrictedparameter estimation and preliminary parameter estimation in linear model, and getsome new biased estimators and results.For the linear model with equality constraints, we study properties of the restrictedleast squares estimator, new restricted ridge estimator, new restricted Liu estimator andnew two-parameter estimator. And based on them, we proposed a new restrictedtwo-parameter estimator which can always satisfy the given linear equality restrictions.We also discuss the performances of the new estimator over other competitiveestimators with respect to mean square error matrix. Furthermore, in order to reduce thebias of a biased estimator, this paper generalize the almost unbiased two-parameterestimator proposed recently in literature to the case with restrictions, namely we havestuded the restricted almost unbiased two-parameter estimator in this paper. Andsimilarly, we prove that the restricted almost unbiased two-parameter is performedbetter than the almost unbiased two-parameter under the mean square error matrixcriterion.For the linear model with stochastic linear constraints, when the prior informationand the sample information are not equally important, by combining the weightedmixed estimator with the almost unbiased ridge estimator and the almost unbiased Liuestimator, respectively, we propose the weighted mixed almost unbiased ridge estimatorand the weighted mixed almost unbiased Liu estimator for unknown parameters in alinear regression model when additional stochastic linear restriction is supposed to beheld. Furthermore, we discuss the performance of the weighted mixed almost unbiasedridge estimator and the weighted mixed almost unbiased Liu estimator with respect tothe quadratic bias and the mean square error matrix criteria and give a method abouthow to obtain the optimal values of parameters k and w.For the multi-normal distribution model when exact linear restrictions are used, wehave firstly discussed some large sample tests widely used in econometric models suchas the Wald test, Likelihood Ratio (LR) test and Lagrangian Multiplier (LM) test. Andthen, we have proposed three preliminary test almost unbiased two-parameter estimatorsbased on the Wald, LR and LM tests when it is suspected that the regression coefficients are the subspace of the equality restrictions. The large sample properties of preliminarytest almost unbiased two-parameter estimators are obtained. And the minimum andmaximum guaranteed efficiency is discussed. In addition, considering the fat-tailphenomenon that may exist in practical data, we have also studed the preliminary testalmost unbiased two-parameter estimator for the multi-t distribution model.Performances of the estimator according to the quadratic bias and mean square error aresimilarly compared in detail.Finnally, when it is suspected that regression coefficients may be restricted to asubspace, we discuss the parameter estimation of regression coefficients in a multipleregression model. Then, in order to improve the preliminary test almost ridge estimator,we study the preliminary test positive-rule stein-type almost unbiased ridge estimatorbased on the positive-rule stein-type shrinkage estimator and almost unbiased ridgeestimator. After that, quadratic bias and quadratic risk values of the new estimator arederived and compared with some relative estimators.