The Large Sample Properties Of The Priestley-Chao Regression Estimation For α-Mixing Sequences

Let Yn1, Yn2,…, Ynn be n observations at fixed xn1,xn2,…, xnn according to the model where the errors{εno} are strong mixing random variables squences with zero mean and finite variance. m is an unknown function defined on x contained in the interval [0,1], and without loss of generality we assume0=xn0≤xn1≤xn2≤…≤xn,n-1≤xnn=1. The Priestley-Chao estimate of m(x) is of the form where K(·) is a measurable function, and{hn} is a sequence of positive real numbers converging to zero as nâ†'∞.For the consistency of Priestley-Chao regression estimation, many scholars have made an intensive study either for independent sample case or for dependent sample case, and obtained a large number of results, but rarely give the convergence rate. For the asymptotic normality, Benedetti(1977) proved a result under independent sample cases, but did not give the convergence rate. Under dependent sample cases, people mainly discussed the asymptotic normality of the general weighted regression function estimate. However, few people studied specially the asymp-totic normality and the convergence rate of Priestley-Chao regression estimation. So, in this paper, we will study the strong consistency and uniformly asymptotic normality of Priestley-Chao ker- nel estimation for α-ixing random sequences. The main research contents and results are as follows:Firstly, we discuss the strong consistency property of Priestley-Chao estimation, and prove the corresponding convergent rates when the random variables are bounded. When the random variables exist r-th order moments (r>2), we use a method differently from Xubing(2002), namely moment inequality, to prove the strong consistency, which makes the proof procedure simpler.Secondly, the uniformly asymptotic normality of mn(x) are discussed by the moment in-equality and the large and small blocks ways. And we obtains the better convergent rate under an appropriate bandwidth. The rate is near n-1/10.Thirdly, we further illustrate the asymptotic properties of the estimate through some numeri-cal simulation.