Research On The Properties Of Parameter Estimation In Linear Regression Model

Linear model is important in statistical model, and has become the most widelyused models in modern statistics. In this paper, we mainly study the properties ofparameter estimation in linear model when prior information is given in the form ofstochastic linear constraints and equality constraints.For the linear model with no constraints, firstly, using the method of almostunbiased estimation, an almost unbiased two-parameter estimator is proposed.Necessary and sufficient conditions for the superiority of the almost unbiasedtwo-parameter estimator over the ordinary least squares estimator and thetwo-parameter estimator in the mean squared error sense are derived. We also give aiterative method to choose the biasing parameters. Secondly, a principal componentLiu-type estimator is introduced by combining the principal component regressionestimator and the Liu-type estimator. The superiority of the principal componentLiu-type estimator over the ordinary least squares estimator, the principal componentregression estimator and the Liu-type estimator are studied in the mean squared errormatrix criterion. Finally, the performance of the r-k class estimator relative to theordinary least squares estimator in the Pitman’s closeness criterion is studied.For the linear model with linear equality constraints, firstly, we compare therestricted ridge estimator and the restricted least squares estimator in the balanced lossfunction. We also give a way to choose the biasing parameter. Secondly, an almostunbiased restricted ridge estimator is proposed which is satisfied the exact linearrestrictions. We also show that under certain conditions, the almost unbiased restrictedridge estimator is superior to the restricted least squares estimator, restricted ridgeestimator and almost unbiased ridge estimator with respective to the mean squared errormatrix sense.For the linear model with stochastic constraints, firstly, a stochastic restrictedtwo-parameter estimator is proposed by combining the two-parameter estimator and themixed estimator. Necessary and sufficient conditions for the superiority of the stochasticrestricted two-parameter estimator over the two-parameter estimator and the mixedestimator with respect to the mean squared error matrix criterion are derived. Secondly,we discuss the property of the weighted mixed estimator. We introduce two relativeefficiencies, and the lower and upper bounds of the two relative efficiencies are also given. At last, we compare the estimators in Panel data model with stochasticrestrictions under the Pitman’s closeness criterion.Finally, we consider the singular linear model. Four new norms of relativeefficiencies are introduced, and the lower bounds of the four relative efficiencies arealso given.