1. Introduction
Let be independent observations of a valued random vector (X, Y) with. We estimate the regression function by the following form of kernel estimates
(1.1)
where is called the 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 can also refer to, for examples, Stone ([3]), Schuster and Yakowitz ([4]), Gasser and Muller ([5]), Mack and Müller ([6]), Greblicki and Pawlak ([7]), 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
(1.2)
and
(1.3)
where, and are positive constants, is either always or always norm, denotes the indicator function of a set, and H is a bounded decreasing Borel function in such that
(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 is a strongly consistent estimate of for almost all (µ probability distribution of X). The first general result in this direction belongs to Devroye ([11]), who established strong pointwise consistency of for bounded Y. Zhao and Fang ([12]) establish its strong consistency under the weaker condition that for some. However, the dominating function of (1.3) in the above literature is confined as for some. GreblickiKrzyżak and Pawlak ([13]) establish the complete convergence of for bounded Y and rather general dominating function H of (1.3) for almost all. This permits to apply kernels with unbounded support and even not integrable ones. In this paper, we establish the strong consistency of 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 Theorem 1.1 Assume that for some, and (1.2)-(1.5) are satisfied, and that
(1.6)
Then
(1.7)
Theorem 1.2 Assume that for some and, and (1.2)-(1.5) are met, and that
(1.8)
Then (1.7) is true.
It is worthwhile to point out that in the above theorems we do not impose any restriction on the probability distribution µ of X.
2. Proof of the Theorems
For simplicity, denote by c a positive constant, by a positive constant depending 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,.
Lemma 2.1 Assume that. For allthere exists a nonnegative function with such that for almost all,
Refer to Devroye ([11]).
Lemma 2.2 Assume that (1.2)-(1.5) are satisfied. Let be integrable for some. Then
as for almost all.
It is easily proved by using Lemma 1 of GKP ([13]).
Lemma 2.3 Assume that (1.2)-(1.5) are met, and that
.
Then for almost all
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”. Write
Since
and by Lemma 2.3, a.s. for a.e. x, it suffices to vertify that a.s. for a.e. x, or, to prove a.s. and a.s. for a.e. x.
Since is convex in y for, and for fixed and, is convex in
for large a, it follows from Jensen’s inequality that and
when, and that
and
for some and
when.
Write, (in Theorem 1.1) or (in Theorem 1.2). It follows that
and
by Borel-Cantelli’s lemma, and
a.s. (2.1)
Write, if. By (1.6) or
(1.8), , we can take such that
(2.2)
Put
By (1.3) and Lemma 2.1, for a.e. x,
(2.3)
By Lemma 2.3,
(2.4)
By Schwarz’s inequality, (2.1), (2.3) and (2.4),
(2.5)
Write
We have Take
. Since for, we have
and
By Lemma 2.2,
.
By (2.2) and (2.3),
Given, it follows that for a.e. x and for n large,
(2.6)
and
By Borel-Cantelli’s lemma and for a.e. x,
for anywe have
a.s for a.e. x Since, by Lemma 2.2, for a.e. x
we have
a.s for a.e. x, as. (2.7)
By (2.2) and (2.3), when, for a.e. x,
and for n large, and
a.s. for a.e. x. (2.8)
By (2.5) and (2.8), noticing that , we have
a.s for a.e. x. (2.9)
To prove a.s for a.e. x, we write , and put
By using the same argument as above,
a.s.
and for a.e. x,
(2.10)
Also, for a.e. x and for n large,
and (2.11)
Write, then
. Take. Since
for, we have
and
By Lemma 2.2, for a.e. x,
Given, similar to (2.6), for a.e. x and n large,
and
and it follows that
a.s. for a.e. x (2.12)
from
for a.e. x and
and Borel-Cantelli’s lemma.
By (2.10)-(2.12),
a.s. for a.e. x and
a.s. for a.e. x (2.13)
Replacing by, it implies that
a.s. for a.e. x (2.14)
(2.13) and (2.14) give
a.s. for a.e. x (2.15)
The theorems follow from (2.9) and (2.15).
3. Acknowledgements
Cui’s research was supported by the Natural Science Foundation of Anhui Province (Grant No.1308085MA02), the National Natural Science Foundation of China (Grant No. 10971210), and the Knowledge Innovation Program of Chinese Academy of Sciences (KJCX3-SYW-S02).