| Nonparametric regression estimation is a very good choice when the population distribution cannot be characterized by a finite number of real parameters and can only make some general assumptions such as continuous distribution,density,and certain moments in the actual data sta-tistical analysis,due to various reasons.The nonparametric regression estimation requires less assumptions and the operation is simpler.It has a wide range of applications in practice,such as the volatility of stock return,stock price forecasting,weather forecast,short-term traffic flow,child growth curve,disease prediction and Artificial Neural Network and so on.Up to now,there are many papers about the estimation.Also many methods for it here,such as kernel estimator,nearest neighbor methods?NN?,partitioning estimate and local polynomial es-timation and so on.One of the most common methods is kernel estimator.The NW kernel weight function is proposed,since the classical nonparametric estimators based on kernel weights have been studied by Nadaraya[2]and Watson[3]?1964?.Later,the more developments of the i.i.d.case and mixing case have been obtained,for instance,by Collomb[4–6]?1976,1977,1981?,Devroye and Wagner[7,8]?1980a,1980b?and Yang[9]?1995?.However,Kernel methods are weighted averages of the response variables and therefore are highly sensitive to large fluctuations in the data.Thus,robust estimation is proposed to solve the weak anti-interference of traditional robust estimates.Therefore,it is of great significance to strengthen the study of robust nonparametric kernel regres-sionHuber,Tukey,Rousseeuw,Fraiman,Cox and other scholars make a fruitful study about robust estimators of the parameters,which Huber[12]?1979?,Huber and Roncheti[13]?2009?,and Maronna et al.[14]?2006?publish some books,since the concept of robustness is firstly proposed by Box in 1953.Zhao and Ma,Boente and Rodriguez and Boente et al.also studied the robust kernel regression estimation of G-M kernel,P-C kernel and NW kernel respectively,and obtained good theoretical properties.In recent years,the researches about NW kernel regression estimator of robust conditional locational function,so that more attention should be paid for it.Boente and Fraiman[21]?1989b?define a robust conditional location function and give the strong consistency and asymptotic nor-mality of NW kernel regression estimator of robust nonparametric in the i.i.d.case.Later,the?-mixing case and?-mixing case can be seen in Boente and Fraiman[1]?1989a?respectively.Im-mediately,Boente and Fraiman[23]?1990?obtain the strong consistency and the asymptotic normal-ity of the NW kernel estimator and the k-nearest neighbor estimator.Based on previous researches,this paper uses more general conditions to study the strong consistency and asymptotic normality of the NW kernel regression estimator of the robust conditional location function for?-mixing case and?-mixing case.First,under certain conditions,we prove that the conditional distribu-tion function sequence converges almost everywhere in the conditional distribution function for?-mixing case and?-mixing case.Then,we give the strong consistency of NW kernel regression estimator of robust nonparametric under certain conditions and finally the asymptotic normality. |
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