Strong Consistency of Kernel Regression Estimate

In this paper, regression function estimation from independent and identically distributed data is considered. We establish strong pointwise consistency of the famous Nadaraya-Watson estimator under weaker conditions which permit to apply kernels with unbounded support and even not integrable ones and provide a general approach for constructing strongly consistent kernel estimates of regression functions.


Introduction
), Kohler, Krzyżak and Walk ([8,9]), and Walk ([10]).When point x is near the boundary of their support, the kernel regression estimator (1.1) has suffered from a serious problem of boundary effects.Hereafter 0/0 is treated as 0. For the kernel function we assume that where 1 2 , and are positive constants, , c c c  is either always 2 or always l l  norm,   This permits to apply kernels with unbounded support and even not integrable ones.In this paper, we establish the strong consistency of   n m x under the conditions of GKP ( [13]) on the kernel and various moment conditions on Y, which provides a general approach for constructing strongly consistent kernel estimates of regression functions.We have It is worthwhile to point out that in the above theorems we do not impose any restriction on the probability distribution µ of X.

Proof of the Theorems
there exists a nonnegative function with It is easily proved by using Lemma 1 of GKP ( [13]).Lemma 2. 3 Assume that (1.2)-(1.5)are met, and that Refer to GKP ( [13]).Now we are in a position to prove Theorems 1.1 and 1.2.
Proof.For simplicity, we write "for a.e.x" instead of the longer phrase "for almost all x R and by Lemma 2.
, and that , we can take such that log ,
(2.8)  By (2.5) and (2.8), noticing that (2.9) , and put (2.10) Also, for a.e.x and for n large, (2.11) and bandwidth and K is a given nonnegative Borel kernel.The estimator (1.1) was first introduced by Nadaraya ([1]) and Watson ([2]).The studies of  m x can also refer to, for examples, Stone ([3]), Schuster and Yakowitz ([4]), Gasser and Muller ([5]), Mack and Müller ([6]), denotes the indicator function of a set, and H is a bounded decreasing Borel function in such that  0,  d (1.4)Through this paper we assume that (1.5)One of the fundamental problems of asymptotic study on nonparametric regression is to find the conditions under which distribution of X).The first general result in this direction belongs toDevroye ([11]), who established strong pointwise consistency of   n m x for bounded Y.Zhao and Fang ([12]) establish its strong consistency under the weaker condition that of (1.3) in the above litera- H x   ture is confined as I x r for some .Greblicki, 0 r   Krzyżak and Pawlak ([13]) establish the complete convergence of   n m x for bounded Y and rather general dominating function H of (1.3) for almost all  

For
simplicity, denote by c a positive constant, by   on x.These constants may assume different values in different places, even within the same expression.We denote by as a sphere of the radius r centered at x, .