Robust And Efficient Estimation For Nonparametric Models And Applications

Regression models generally include parametric and nonparametric regression models,where parametric regression models usually assume that the form of the regression function between variables is known,and only the unknown parameters need to be estimated.However,when the relationship between response and explanatory variables can not be determined,the parametric regression models may not work well in fitting the data,and nonparametric regression models can make up for this deficiency.The relationship between response and explanatory variables in nonparametric regression models does not take a predetermined form but is constructed according to information derived from the data,then it makes nonparametric models more adaptable.In the past decades,nonparametric regression models have been widely used in social,medical,biological,psychological and educational fields,but the research on nonparametric models in the complex environment need to be improved.This dissertation primarily studies the problems of robust estimation,bandwidth selection and applications for nonparametric models in different situations,including a robust estimation of nonparametric models with mixed discrete and continuous data,a robust estimation of nonparametric models with jump points,a robust estimation of the link function with jump points in single index models and the kernel estimation based on Bayesian bandwidth selection in nonparametric models.The main contents are arranged as follows:Chapter 1 is devoted to introducing the research background,significance,status and the existing problems for nonparametric models.In addition,an outline of this dissertation about our major work is given,encompassing innovative keys.Chapter 2 focuses on a robust and efficient estimation for nonparametric models with mixed variables.Considering continuous and categorical variables,a robust and efficient estimation method is proposed to estimate the regression function.Under some mild conditions,the asymptomatic properties of the resulting estimators are established.Furthermore,several numerical studies are conducted to evaluate the finite sample performance of the proposed methodologies.Finally,an application with real data illustrates the usefulness of the proposed techniques.Chapter 3 is devoted to studying the estimation for nonparametric models with unknown jump points.In practical applications,using observations to estimate regression function directly may be misleading,because regression function is discontinuous in some unknown positions.Thus,how to deal with unknown jump points in the regression function estimation is important.Firstly,a robust method is proposed to estimate regression function based on the one-side kernel function.Then,since the local linear estimation procedure inevitably has the huge computation burden,an alternative method based on B-spline is proposed to detect jump points.Furthermore,some other methods are considered to reflect the superiority of the proposed methods.Results of Monte Carlo experiments are presented to examine the finite sample performances of the proposed procedures and real data analysis are used to verify the feasibility of the proposed method.Chapter 4 is concerned with a robust estimation for the link function with jumps in single index models.Based on the one-side kernel function,a robust and efficient method is proposed for the single index models with discontinuous link function.The proposed estimator can overcome the curse of dimensionality and is shown to be robust to the presence of outlier or heavy-tailed distributions.Under some mild conditions,the consistency and asymptotic normality for these estimators are studied explicitly.Moreover,some simulation studies are carried out to examine the finite sample performance of the proposed method,and a real data analysis illustrates the feasibility of the proposed method.Chapter 5 investigates the kernel estimation method based on Bayesian bandwidth selection in nonparametric models.Using distribution information of the error item,a Bayesian bandwidth selection method is proposed for nonparametric models.The proposed method provides estimators for posterior bandwidth respectively when the error distribution is known or unknown,and the large sample properties are also established.The finite sample performance of the proposed procedures is assessed by Monte Carlo simulation experiments and a real data set example.