Partial Linear Quantile Regression Via An MM Algorithm

Semiparametric quantile regression model is a kind of very important model which has the advantage of processing data's heavy tail traits and outliers and catching data's quantile structure.In addition,it can depict linear or nonlinear relationship between variables and overcome the problem of "dimension curse”to some degree.In view of these,semiparametric quantile regression model has gradually become a research hotspot in the field of regression analysis,among which the partial linear quantile regression model is the most widely used.Partial linear quantile regression model has no assumption of homogeneity and normality for error terms so that the traditional estimation methods of mean reversion are no longer applicable.To solve this problem,two-stage estimation method is presented.At present,the effective and generally accepted calculation methods include simplex method,interior point method and pretreatment interior point method.However,when we deal with massive or high-dimensional data,the above methods show weakness of unstable numerical calculation and low computational speed.In 2000,Hunter and Lange proposed the MM algorithm which has been proved numerical stability.In order to estimate the partial linear quantile regression model,we propose a new MM algorithm based on kernel estimation which can overcome the problem of multiple minimum points caused by the unsmoothness of the objective function.The main principle is to construct the optimization function of the objective function and iterate to the solution of the objective function by means of the minimization process of the optimization function.In this paper,we elaborate on the new algorithm's steps and prove that the approximate solution obtained by the new algorithm converges to the solution of the original objective function.Furthermore,we determine the upper bound of the distance between the minimum of the approximate function and the minimum of the original objective function.Finally,the experimental results of the numerical simulation and real data analysis show that the proposed algorithm provides an effective way to solve this type of model,overcome the interference of non-normal error and improve the estimation efficiency significantly.