Research On Several Biased Estimators In Linear Models

When there is complex collinearity present in a linear model,the least squares estimation may inflate,and the signs and actual conditions may not match.This paper proposes several new types of biased estimations based on principal component estimation,KL estimation,and modified KL estimation method in linear models with an equality constraint.The proposed methods aim to overcome the influence of complex collinearity.The specific content of the paper includes:Firstly,in linear models without an equality constraint,this paper proposes KL-type principal component estimation and modified KL-type principal component estimation by combining principal component estimation with KL estimation and modified KL estimation.The sufficient and necessary conditions for the optimality of KL-type principal component estimation over principal component estimation,KL estimation,etc.,in terms of mean square error,are obtained.Moreover,the sufficient and necessary conditions for the optimality of modified KL-type principal component estimation over principal component estimation,modified KL estimation,KL-type principal component estimation,etc.,are derived.In addition,Monte Carlo simulation method and examples are used to validate and demonstrate the theoretical results.Secondly,for linear models with an equality constraint,this paper proposes constrained KL-type principal component estimation and constrained modified KL-type principal component estimation based on constrained least squares estimation method.The sufficient and necessary conditions for the optimality of constrained KL-type principal component estimation over KL-type principal component estimation,constrained least squares estimation,etc.,in terms of the mean square error matrix,are obtained.Moreover,the sufficient and necessary conditions for the optimality of constrained modified KL-type principal component estimation over modified KL-type principal component estimation,constrained least squares estimation,etc.,are derived.Additionally,the performance of the two types of new estimations is demonstrated through Monte Carlo simulation and examples analysis in terms of mean square error.Finally,a summary of this study is presented.Although this article has addressed some significant practical issues,there are still some limitations that need to be addressed.Additionally,future research directions are proposed to further explore these issues.