The Large Sample Properties Of Gasser-Müller Estimation For -Mixing Sequences

The nonparametric regression estimation was formed in 1970s, and developed gradually. In recent 20 years, the method is widely used in econometric models, financial asset price and volatil-ity of return rate. However, in nonparametric regression estimation, the weighted estimator of regression function is always employed. In 1977, the weighted function of estimation method pro-posed by Stone which is belonged to nonparametric, has been widely recognized by scholars. For the design regression modelYj= g(xj)+εj,1≤j≤n, Gasser and Muller (1979) introduced the weight function Where K(·) is Borel measurable function, and 0<hn�'40(when n�'∞).In 1990, Bradly, R, C proposed the concept of ρ-mixing sequence, and it played an irre-placeable role in many areas of social sciences, natural sciences, engineering and technology, for example, we can use the convergence of ρ-mixing sequence to analyse the stock. Today, many scholars study the statistical properties of the general weight function regression in the ρ-mixing sequence. However, for the specific weighted function, such as Gasser-muller, there are not too many people to study it. So in this paper, it is meaningful to study the complete consistency, the strong consistency and the asymptotic normality of Gasser-muller regression estimation. The main research contents and results are as follows:Firstly, we study the complete consistency of Gasser-muller regression estimation based on ρ-mixing sequence, by using Rosenthal-type inequality and the truncated method. The main different between this paper and Wang(2012)is the truncation, in this paper, we don’t have to consider the weight ωnj (x), just to truncate the ρ-mixing sequence at |εj|= nr-1Secondly, we discuss the strong consistency of the Gasser-muller regression based on p-mixing sequence. We use the subsequence method and theorem Kronecker in deal with the end of this sequence, which makes the processes of proofing easily.Thirdly, we study the asymptotic normality of Gasse-muller regression weighted estimation for p-mixing sequence case by lemma Lyapunor and the large and small blocks ways.Finally, we can generate a p-mixing sequence through the time series model of MA(1), so we can further illustrate the asymptotic properties of the Gasser-muller regression weighted estimation through some numerical simulation.