Two-stage Estimation Of Regression Parameters For Linear Model

The linear model can approximately describe phenomena in the fields of biolo-gy,medicine,economy,management,geology,meteorology,agriculture,industry,en-gineering technology,etc.It is the most widely used in modern statistics.Under the assumption that the error covariance matrix is?²I_n,the least squares estimation method is usually used to estimate the regression parameters of the linear model.But in many cases,the error covariance matrix of the linear model has the form?²?,and?often contains unknown parameters.According to Gauss–Markov theorem,the generalized least squares estimate is the best linear unbiased estimate,and the generalized least squares estimate is better than the least squares estimate.However,because the gen-eralized least squares estimation contains unknown parameters,the generalized least squares estimation loses the application conditions.The article uses a two-stage estimation method to estimate the regression parame-ters.It mainly studies the two-stage estimation of the regression parameters of a class of linear model with special error covariance structures.The current research results on two-stage estimation are remarkable.For example,the two-stage estimation in the Panel data model^[1][2],the two-stage estimation of the growth curve model^[3],regres-sion System generalized ridge principal component two-stage estimation^[4],two-stage estimation of heteroscedastic structure model^[11],and two-stage estimation of circle parameters^[12],the two-stage estimate of the regression equation system^[13]and so on.In these model structures,the two-stage estimation exhibits excellent characteristics.In this article,the first step gives the expression of generalized least squares esti-mation,and discusses the conditions under which generalized least squares estimation is equal to the least squares estimation;then,on this basis,the parameters of?are es-timated and obtained one expression,which verifies the unbiasedness of the parameter estimate,and it is an even function of the observation vector and has transformation invariance;finally,the superiority of the two-stage estimation is proved,and the expres-sion for evaluating the efficiency of the two-stage estimation is studied and the condition that the two-stage estimation is better than the least square estimation is gived.Finally,the article uses Monte Carlo simulation to give the parameter estimation of?and the numerical simulation of the two-stage estimation of regression parameters.During the simulation,the real value of the parameter?is changed between[-0.5,0.5],and then the generalized least squares estimation is available.Through the simulation results,it is found that when the sample size is large,the mean square error of the two-stage estimate is significantly smaller than the least squares estimate;and,as the sample size continues to increase,the mean square error of the two-stage estimate continues to approach the generalized minimum square estimation.So,the two-stage estimation is better than the least squares estimation.The actual data of the coal purification process is used at the end of the article to further illustrate the superiority of the two-stage estimation over the least squares estimation.