Complete Consistency of the Estimator of Nonparametric Regression Models Based on -Mixing Sequences
We study the complete consistency for estimator of nonparametric regression model based on -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.
Consider the following fixed design nonparametric regression model: where are known fixed design points from , where is a given compact set for some , is an unknown regression function defined on , and are random errors. Assume that for each , have the same distribution as . As an estimator of , the following weighted regression estimator will be considered: where , are the weight function.
The above estimator was first proposed by Georgiev  and subsequently has been studied by many authors. For instance, when are assumed to be independent, consistency and asymptotic normality have been studied by Georgiev and Greblicki , Georgiev  and Müller  among others. Results for the case when are dependent have also been studied by various authors in recent years. Fan  extended the work of Georgiev  and Müller  in the estimation of the regression model to the case that forms an -mixingale sequence for some . Roussas  discussed strong consistency and quadratic mean consistency for under mixing conditions. Roussas et al.  established asymptotic normality of assuming that the errors are from a strictly stationary stochastic process and satisfying the strong mixing condition. Tran et al.  discussed again asymptotic normality of 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.  studied the asymptotic normality for double array sum of linear time series. Hu et al.  gave the mean consistency, complete consistency, and asymptotic normality of regression models with linear process errors. Liang and Jing  presented some asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences, Yang et al.  generalized the results of Liang and Jing  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 -mixing random variables.
In the following, we will give the definition of sequence of -mixing random variables.
Let be a random variable sequence defined on a fixed probability space . Write . Given -algebras in , let Define the -mixing coefficients by Obviously, , and .
Definition 1.1. A sequence of random variables is said to be a -mixing sequence if there exists such that .
-mixing random variables were introduced by Bradley , and many applications have been found. Many authors have studied this concept providing interesting results and applications. See for example, Bradley  for the central limit theorem, Bryc and Smoleński , Peligrad , and Utev and Peligrad  for moment inequalities, Gan , Kuczmaszewska , Wu and Jiang , and Wang et al.  for almost sure convergence, Peligrad and Gut , Gan , Cai , Kuczmaszewska , Zhu , An and Yuan , Sung  and Wang et al.  for complete convergence, and Peligrad  for invariance principle, Zhou et al.  and Sung  for strong law of large numbers, and so forth. When these are compared with the corresponding results of independent random variable sequences, there still remains much to be desired.
Throughout the paper, denotes a positive constant not depending on , which may be different in various places. represents for all . Let denote the integer part of and let be the indicator function of the set . Denote and .
2. Main Result
Unless otherwise specified, we assume throughout the paper that is defined by (1.2). For any function , we use to denote all continuity points of the function on . The norm is the Euclidean norm. For any fixed design point , the following assumptions on weight function will be used:) as ;() for all ;() as for all .
Based on the assumptions above, we can get the following complete consistency of the nonparametric regression estimator .
Theorem 2.1. Let be a sequence of -mixing random variables with mean zero, which is stochastically dominated by a random variable . Assume that conditions – hold true. If there exists some such that and then
As an application of Theorem 2.1, we give the complete consistency for the nearest neighbor estimator of . Without loss of generality, put , taking , . For any , we rewrite as follows: if , then is permuted before when .
Let , the nearest neighbor weight function estimator of in model (1.1) is defined as follows: where
Based on the notations above, we can get the following result by using Theorem 2.1.
Corollary 2.2. Let be a sequence of -mixing random variables with mean zero, which is stochastically dominated by a random variable . Suppose that is continuous on the compact set . If there exists some such that and , then
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 .
Lemma 3.1. Let be a -mixing sequence of random variables, , for some and for every . Then there exists a positive constant C depending only on such that
The concept of stochastic domination will be used in this work.
Definition 3.2. A sequence of random variables is said to be stochastically dominated by a random variable if there exists a positive constant such that for all and .
Lemma 3.3. Let be a sequence of random variables which is stochastically dominated by a random variable . For any and , the following two statements hold: where and are positive constants.
4. Proofs of the Main Results
Proof of Theorem 2.1. For and , we have by (1.1) and (1.2) that
Since , hence for any , there exists a such that when . Thus, by setting in (4.1), we can get that
By conditions ()–() and the arbitrariness of , we can get that
For fixed design point , note that , so without loss of generality, we assume that in what follows.
From the condition (2.1), we assume that 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 , For fixed , denote It is easy to check that for any , which implies that Hence, to prove (4.6), it suffices to show that and .
By condition () and , we can get that which implies that .
Next, we will prove that . Firstly, we will show that Actually, by the conditions , Lemma 3.3, (4.4) and , we can see that which implies (4.11). Hence, to prove , we only need to show that for all , By Markov’s inequality, Lemma 3.1, ’s inequality, and Jensen’s inequality, we have for that Take which implies that and . By ’s inequality and Lemma 3.3, we can get If , then we have by Markov’s inequality, and (4.4) that If , then we have by Markov’s inequality, and (4.4) again that From (4.14)–(4.18), we have proved that .
By ’s inequality and Lemma 3.3, we can see that has been proved by (4.10). In the following, we will show that . Denote It is easily seen that for and for all . Hence, It is easily seen that for all , which implies that for all , Therefore, Thus, the inequality (4.13) follows from (4.14)–(4.19), (4.21), and (4.24). This completes the proof of the theorem.
Proof of Corollary 2.2. It suffices to show that the conditions of Theorem 2.1 are satisfied. Since is continuous on the compact set , hence, is uniformly continuous on the compact set , which implies that is bounded on set .
For any , if follows from the definition of and that Hence, conditions ()–() and (2.1) are satisfied. By Theorem 2.1, we can get (2.6) immediately. This completes the proof of the corollary.
This paper is supported by the National Natural Science Foundation of China (11201001, 11171001, and 11126176), Natural Science Foundation of Anhui Province (1208085QA03), Doctoral Research Start-up Funds Projects of Anhui University, Academic Innovation Team of Anhui University (KJTD001B), University Students Science Research Training Program of Anhui University (KYXL2012007), and Students’ Innovative Training Project of Anhui University (cxcy2012003). The authors are most grateful to the editor Ciprian A. Tudor and anonymous referee for careful reading of the paper and valuable suggestions which helped in significantly improving an earlier version of this paper.
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