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 all
there exists a nonnegative function
with
such that for almost all
,
![](https://www.scirp.org/html/3-1240183\f69262f8-684b-4657-8dc1-b09ffde780c9.jpg)
Refer to Devroye ([11]).
Lemma 2.2 Assume that (1.2)-(1.5) are satisfied. Let
be
integrable for some
. Then
![](https://www.scirp.org/html/3-1240183\24e5b6c3-ca32-4989-8820-6c75e96b157a.jpg)
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![](https://www.scirp.org/html/3-1240183\33685bb4-8a67-4188-8dc8-8e66c13782d2.jpg)
![](https://www.scirp.org/html/3-1240183\325984a3-b1f9-40cb-b074-3277e80dd8dd.jpg)
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
![](https://www.scirp.org/html/3-1240183\8ec3d4cd-2c78-4307-a741-fd1cd6831024.jpg)
Since
![](https://www.scirp.org/html/3-1240183\fa2016a6-39a0-4b80-b6ce-e395ef75d03a.jpg)
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 ![](https://www.scirp.org/html/3-1240183\d1597a04-05fb-4c46-b7fa-9e83cf801ed1.jpg)
when
, and that
and
for some
and ![](https://www.scirp.org/html/3-1240183\ca5bd184-5605-47f0-a1c8-dd4fba5ac4b4.jpg)
when
.
Write
,
(in Theorem 1.1) or
(in Theorem 1.2). It follows that
![](https://www.scirp.org/html/3-1240183\49953171-6134-45b0-97be-76ced0ccc43a.jpg)
and
![](https://www.scirp.org/html/3-1240183\aabc7df5-e3fb-45a8-be90-6b1d41cc76f9.jpg)
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
![](https://www.scirp.org/html/3-1240183\5268636f-3df4-4ea7-9bdf-670bf7218a80.jpg)
![](https://www.scirp.org/html/3-1240183\52f32a89-42e1-4f12-a140-c03fe93013d3.jpg)
![](https://www.scirp.org/html/3-1240183\bd45ec6c-ebec-4998-8277-5f17b7ab9554.jpg)
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
![](https://www.scirp.org/html/3-1240183\621f5203-45de-4cb2-b29c-e83b1f93e6b4.jpg)
We have
Take
. Since
for
, we have
![](https://www.scirp.org/html/3-1240183\c5df96ef-3f33-465a-8699-6e18370740d4.jpg)
and
![](https://www.scirp.org/html/3-1240183\dfd87d01-1e5a-40f1-893e-2e32f936fbe6.jpg)
By Lemma 2.2,
.
By (2.2) and (2.3),
![](https://www.scirp.org/html/3-1240183\ab75e1db-c4bf-42d2-9851-b8658dc8b3f5.jpg)
Given
, it follows that for a.e. x and for n large,
(2.6)
and
![](https://www.scirp.org/html/3-1240183\5bfef55a-ea2d-4963-90cc-4ed0b5dff564.jpg)
By Borel-Cantelli’s lemma and for a.e. x,
for any
we have
a.s for a.e. x Since, by Lemma 2.2, for a.e. x
![](https://www.scirp.org/html/3-1240183\3061da83-aa19-4bb7-825e-d9577e215ec2.jpg)
we have
a.s for a.e. x, as
. (2.7)
By (2.2) and (2.3), when
, for a.e. x,
![](https://www.scirp.org/html/3-1240183\19356636-435e-46fb-8433-4de5a43653db.jpg)
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
![](https://www.scirp.org/html/3-1240183\172aa9f8-31c8-4969-b249-bf37183a400e.jpg)
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
![](https://www.scirp.org/html/3-1240183\4b9bfe36-7be6-4d0c-8da4-a3c6d619eb02.jpg)
and
![](https://www.scirp.org/html/3-1240183\86a4d420-dda3-4ffd-bdab-29ad9027c9c2.jpg)
By Lemma 2.2, for a.e. x,
![](https://www.scirp.org/html/3-1240183\5358d521-5b5b-4081-bf1c-e4a14b187a5f.jpg)
Given
, similar to (2.6), for a.e. x and n large,
and
![](https://www.scirp.org/html/3-1240183\0c0fa2d5-ed48-4161-9ee1-f4a068d03947.jpg)
and it follows that
a.s. for a.e. x (2.12)
from
for a.e. x and ![](https://www.scirp.org/html/3-1240183\7f2e1ee6-9d09-4156-bdfe-dae0e74042f9.jpg)
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).