The Large Sample Properties Of Nadaraya-Waston Estimation For -Mixing Sequences

Nonparametric statistics is one of the most important branch of statisttics. Consider the fixed design regression model where the design points x1,x2,…,xn ∈ (0,1), A is a compact set of R1, g is a bounded real valued function on A,{εi} are regression errors with zero mean and finite variance.The weighted function estimates of g is The Nadaraya-Waston estimate of weight function is the form where the K (·) is a measurable function, and the {hn} is the window width which is a sequence of positive real numbers covering to zero as n �'∞.For the fixed-design, it has been studied by many schollars in independent case, such as Priestly and Chao[1972], Clark[1977], Grorgiev[1984a,1988], Georgiev and Greblicki[1986]. In various dependent cases, gn(x) has also been investigated extensively.For instance, Fan[1990], Roussas[1989], Roussas et al.[1992],Tran et al[1986], Shanchao Yang[1999] and the references therein. Ioannides[1992]、Rrassas [1992] and Tran[1992] have studied the asymptotic normality of gn(x) under the a condition. Shanchao Yang[2003] has studied the uniformly asymptotic nor-malite of gn(x) under the nagative associated condition. However, there has not been a mature the-ory in ρ-mixing sequence. Thus, we will discuss the large sample properties of Nadaraya-Waston estimate for ρ-mixing sequences. The main research contents and results are as follows:Firstly, we discuss the complete consistency property and strong consistency property of Nadaraya-Waston estimation under p mixing sequences by truncation method and moment in-equality. What is different from Wang[2012] is that we choose the truncation at|εi|= n-1/r-1 of the sequence{εi}.We avoid the trouble of dealing with the wight. Besides, the condition max1≤i≤n Wni(x)=O(n-1/r-1) in this paper is better than Wang[2012].Secondly, we discuss the strong consistency property of Nadaraya-Waston estimation under ρ mixing sequences. We tactfully use subsequence method and Kronecker lemma as dealing wiht the tail.Thirdly, we use the traditional method to prove the asymptotic normality of Nadaraya-Waston estimation under p mixing sequences. Where we divide Sn into Sn’, Sn’’ and Sn’’’, we prove the theorem by Lyapunov central limit theorem.At last, we simulate the strong consistency property and the asymptotic normality of Nadaraya-Waston estimation by the ρ mixing sequences which is generated by MA(1) model.We further explain the theoretical significance and the practical significance of researching the large sample properties of Nadaraya-Waston estimation from the average error and images.