Error Density Estimation Based On Local Polynomial Regression

The density function of error term is often involved in the diagnosis of regression model and the estimation of error variance.When the density function of error term is unknown,we need to estimate it and hope that the estimation has better convergence.While in practical research,we usually can not get information about the error term,but the residual term contains part of the information of the response variable.We can estimate the properties of the error term by studying the residual term.Some scholars have studied error density estimation based on residuals.The common methods are kernel estimation method and frequency histogram method.However,these methods have boundary effect and can not study the properties of the derivatives of error density function.The local polynomial method in nonparametric regression has the remarkable characteristic that there is no boundary effect problem,besides it can estimate the regression function and its derivative function at the same time.At present,this method has important application value in various disciplines.In view of the important role of the error density function and the excellent characteristics of the local polynomial estimation method,a new research theory is proposed in this paper,which constructs the regression error density estimation based on residual as a nonparametric regression model,and uses the local polynomial method to study.Compared with other methods,the method in this paper can solve the problems existing in previous studies,and the rationality and effectiveness of our estimation method are verified by theoretical proof and numerical simulation results.Therefore,this paper is a powerful supplement to the theory of error density.In the first chapter,we first introduce the research background and the development of the regression model,then analyse the significance about the related properties of the error term to the practical research,finally the research status of this paper at home and abroad is briefly summarized;In the second chapter,the models,estimation methods and basic assumptions involved in this paper are introduced,and the main research conclusions are given;In the third chapter,starting from the method of Cattaneo et al.,and combining their research conclusions with the method of this paper,we give the proof of five lemmas and four theorems.According to the obtained theorems,we theoretically prove the asymptotic property and convergence of the method used in this paper;In the fourth chapter,we use our method to simulate the error terms from different distributions.The simulation results show that our method has strong applicability and effectiveness;In the fifth chapter,we discuss the advantages and disadvantages of the research method,and the future research direction.