Biased Estimation Of Parameters For Singular Linear Models With Restrictions

The singular linear model is a special kind of statistical model,and its covariance matrix is singular.Because it has a wide range of applications,this model has also attracted great attention from statisticians.This paper studies the multicollinearity problem of singular linear models.Three new biased estimates of parameters are proposed and their related properties are discussed.Firstly,for the singular linear model,the method of biased estimation is used to unify the universal ridge estimation and Stein estimates into a larger estimation class.Under the mean square error matrix criterion,the sufficient conditions for its advantage over the least squares estimation are discussed.Under the mean square error criterion,it is proved that the Stein-type compression technique can improve the universal ridge estimation,and the universal ridge regression technique can be used to improve the Stein estimation.The numerical simulation is used to verify the superiority of the new estimation.Secondly,for the singular linear model with equality restrictions,an equality-restricted two-parameter estimation is proposed by simulating the two-parameter estimation in the linear model.Under the mean square error matrix criterion,the necessary and sufficient conditions for the new estimation to be better than the restricted least squares estimation,the equality-restricted ridge estimation,and the equality-restricted Liu estimation are obtained.The related theoretical results is verified by numerical simulation.Finally,for the singular linear model with stochastic restrictions,a weighted mixed two-parameter estimation is proposed by adding a new two-parameter estimation operator to the weighted mixed estimation.Under the mean square error matrix criterion,the necessary and sufficient conditions for the new estimation to be better than the weighted mixed estimation,the weighted mixed ridge estimation,the weighted mixed Liu estimation,and the two-parameter estimation are obtained.The related theoretical results is verified by numerical simulation.