Complete Consistency of the Estimator of Nonparametric Regression Models Based on ∼ ρ -Mixing Sequences

We study the complete consistency for estimator of nonparametric regression model based on ∼ ρ mixing sequences by using the classical Rosenthal-type inequality and the truncated method. As an application, the complete consistency for the nearest neighbor estimator is obtained. normality


Introduction
Consider the following fixed design nonparametric regression model: where x ni are known fixed design points from A, where A ⊂ R p is a given compact set for some p ≥ 1, g • is an unknown regression function defined on A, and ε ni are random errors.Assume that for each n ≥ 1, ε n1 , ε n2 , . . ., ε nn have the same distribution as ε 1 , ε 2 , . . ., ε n .As an estimator of g • , the following weighted regression estimator will be considered: where W ni x W ni x; x n1 , x n2 , . . ., x nn , i 1, 2, . . ., n are the weight function.

Abstract and Applied Analysis
The above estimator was first proposed by Georgiev 1 and subsequently has been studied by many authors.For instance, when ε ni are assumed to be independent, consistency and asymptotic normality have been studied by Georgiev and Greblicki 2 , Georgiev 3 and M üller 4 among others.Results for the case when ε ni are dependent have also been studied by various authors in recent years.Fan 5 extended the work of Georgiev 3 and M üller 4 in the estimation of the regression model to the case that forms an L q -mixingale sequence for some 1 ≤ q ≤ 2. Roussas 6 discussed strong consistency and quadratic mean consistency for g n x under mixing conditions.Roussas et al. 7 established asymptotic normality of g n x assuming that the errors are from a strictly stationary stochastic process and satisfying the strong mixing condition.Tran et al. 8 discussed again asymptotic normality of g n x assuming that the errors form a linear time series, more precisely, a weakly stationary linear process based on a martingale difference sequence.Hu et al. 9 studied the asymptotic normality for double array sum of linear time series.Hu et al. 10 gave the mean consistency, complete consistency, and asymptotic normality of regression models with linear process errors.Liang and Jing 11 presented some asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences, Yang et al. 12 generalized the results of Liang and Jing 11 for negatively associated sequences to the case of negatively orthant dependent sequences, and so forth.The main purpose of this section is to investigate the complete consistency for estimator of the nonparametric regression model based on ρ-mixing random variables.
In the following, we will give the definition of sequence of ρ-mixing random variables.Let {X n , n ≥ 1} be a random variable sequence defined on a fixed probability space Ω, F, P .Write When these are compared with the corresponding results of independent random variable sequences, there still remains much to be desired.This work is organized as follows: main result of the paper is provided in Section 2. Some preliminary lemmas are presented in Section 3, and the proof of the main result is given in Section 4.
Throughout the paper, C denotes a positive constant not depending on n, which may be different in various places.a n O b n represents a n ≤ Cb n for all n ≥ 1.Let x denote the integer part of x and let I A be the indicator function of the set A. Denote x xI x ≥ 0 and x − −xI x < 0 .

Main Result
Unless otherwise specified, we assume throughout the paper that g n x is defined by 1.2 .For any function g x , we use c g to denote all continuity points of the function g on A. The norm x is the Euclidean norm.For any fixed design point x ∈ A, the following assumptions on weight function W ni x will be used: Based on the assumptions above, we can get the following complete consistency of the nonparametric regression estimator g n x .
Theorem 2.1.Let {ε n , n ≥ 1} be a sequence of ρ-mixing random variables with mean zero, which is stochastically dominated by a random variable X. Assume that conditions (A 1 ) − (A 3 ) hold true.If there exists some s > 0 such that E|X| 1 1/s < ∞ and As an application of Theorem 2.1, we give the complete consistency for the nearest neighbor estimator of g x .Without loss of generality, put A 0, 1 , taking Let 1 ≤ k n ≤ n, the nearest neighbor weight function estimator of g x in model 1.1 is defined as follows: where 0, otherwise.

2.5
Based on the notations above, we can get the following result by using Theorem 2.1.
Corollary 2.2.Let {ε n , n ≥ 1} be a sequence of ρ-mixing random variables with mean zero, which is stochastically dominated by a random variable X. Suppose that g is continuous on the compact set A.
If there exists some 0 < s < 1 such that k n n s and E|X| 1 1/s < ∞, then

Preliminaries
In this section, we will present some important lemmas which will be used to prove the main results of the paper.The first one is the Rosenthal-type inequality, which was proved by Utev and Peligrad 16 .
Lemma 3.1.Let {X n , n ≥ 1} be a ρ-mixing sequence of random variables, EX i 0, E|X i | p < ∞ for some p ≥ 2 and for every i ≥ 1.Then there exists a positive constant C depending only on p such that The concept of stochastic domination will be used in this work.
Definition 3.2.A sequence {X n , n ≥ 1} of random variables is said to be stochastically dominated by a random variable X if there exists a positive constant C such that for all x ≥ 0 and n ≥ 1.
Lemma 3.3.Let {X n , n ≥ 1} be a sequence of random variables which is stochastically dominated by a random variable X.For any α > 0 and b > 0, the following two statements hold:

3.3
where C 1 and C 2 are positive constants.

Proofs of the Main Results
Proof of Theorem 2.1.For x ∈ c g and a > 0, we have by 1.1 and 1.2 that

4.1
Since x ∈ c g , hence for any ε > 0, there exists a δ > 0 such that |g x − g x | < ε when x − x < δ.Thus, by setting 0 < a < δ in 4.1 , we can get that

4.2
By conditions A 1 -A 3 and the arbitrariness of ε > 0, we can get that lim n → ∞ Eg n x g x , x ∈ c g .

4.3
For fixed design point x ∈ c g , note that W ni x W ni x − W − ni x , so without loss of generality, we assume that W ni x ≥ 0 in what follows.
From the condition 2.1 , we assume that max 1≤i≤n By 4.3 , we can see that in order to prove 2.2 , we only need to show that That is to say, it suffices to show that for all ε > 0, Abstract and Applied Analysis For fixed n ≥ 1, denote It is easy to check that for any ε > 0, which implies that

4.9
Hence, to prove 4.6 , it suffices to show that I < ∞ and J < ∞.By condition A 1 and E|X| 1 1/s < ∞, we can get that which implies that I < ∞.
Next, we will prove that J < ∞.Firstly, we will show that Actually, by the conditions Eε i 0, Lemma 3.3, 4.4 and E|X| 1 1/s < ∞, we can see that 12 which implies 4.11 .Hence, to prove J < ∞, we only need to show that for all ε > 0, By Markov's inequality, Lemma 3.1, C r 's inequality, and Jensen's inequality, we have for p ≥ 2 that

4.14
Take which implies that −sp/2 < −1 and −s p − 1 < −1.By C r 's inequality and Lemma 3.3, we can get

4.18
From 4.14 -4.18 , we have proved that J 1 < ∞.By C r 's inequality and Lemma 3.3, we can see that

4.19
Abstract and Applied Analysis 9 J 3 < ∞ has been proved by 4.10 .In the following, we will show that J 4 < ∞.Denote It is easily seen that I nk I nj ∅ for k / j and ∞ j 1 I nj {1, 2, . . ., n} for all n ≥ 1.Hence,

4.21
It is easily seen that for all m ≥ 1,

4.22
which implies that for all m ≥ 1, Abstract and Applied Analysis Therefore,
Wu and Jiang 19 , and Wang et al. 20 for almost sure convergence, Peligrad and Gut 21 , Gan 17 , Cai 22 , Kuczmaszewska 23 , Zhu 24 , An and Yuan 25 , Sung 26 and Wang et al. 27 for complete convergence, and Peligrad 15 for invariance principle, Zhou et al. 28 and Sung 29 for strong law of large numbers, and so forth.
14 -4.19 , 4.21 , and 4.24.This completes the proof of the theorem.Abstract and Applied Analysis 11For any x ∈ 0, 1 , if follows from the definition of R i x and W ni x thatW ni x • g x ni − g x I |x ni − x| > a ≤ CHence, conditions A 1 -A 3 and 2.1 are satisfied.By Theorem 2.1, we can get 2.6 immediately.This completes the proof of the corollary.
Proof of Corollary 2.2.It suffices to show that the conditions of Theorem 2.1 are satisfied.Since g is continuous on the compact set A, hence, g is uniformly continuous on the compact set A, which implies that {|g x ni − g x | : 1 ≤ i ≤ n, n ≥ 1} is bounded on set A.