ABSTRACT

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.

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).

REFERENCES