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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.

Let be independent observations of a valued random vector (X, Y) with. We estimate the regression function by the following form of kernel estimates

where is called the bandwidth and K is a given nonnegative Borel kernel. The estimator (1.1) was first introduced by Nadaraya ([

and

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

Through this paper we assume that

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 ([

Then

Theorem 1.2 Assume that for some and, and (1.2)-(1.5) are met, and that

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.

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 ([

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 ([

Lemma 2.3 Assume that (1.2)-(1.5) are met, and that

.

Then for almost all

Refer to GKP ([

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

Put

By (1.3) and Lemma 2.1, for a.e. x,

By Lemma 2.3,

By Schwarz’s inequality, (2.1), (2.3) and (2.4),

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,

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,

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).

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).