The Strong Consistency of the Estimator of Fixed-Design Regression Model under Negatively Dependent Sequences
We study the strong consistency of estimator of fixed design regression model under negatively dependent sequences by using the classical Rosenthal-type inequality and the truncated method. As an application, the strong consistency for the nearest neighbor estimator is obtained.
Let be a sequence of random variables defined on a fixed probability space . It is well known that the Rosenthal-type inequality for the partial sum plays an important role in probability limit theory and mathematical statistics. The main purpose of the paper is to investigate the strong consistency of the estimator of fixed design regression model under negatively dependent sequences, by using the Rosenthal-type inequality.
Consider the following fixed design 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 which form 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. Shen  presented the Bernstein-type inequality for widely dependent random variables and gave its application to nonparametric regression models, and so forth. The main purpose of this section is to investigate the strong consistency of estimator of the fixed design regression model based on negatively dependent random variables.
The concept of negatively dependent random variables was introduced by Lehmann  as follows.
A finite collection of random variables is said to be negatively dependent (or negatively orthant dependent, ND in short) if hold for all . An infinite sequence is said to be ND if every finite subcollection is ND.
Obviously, independent random variables are ND. Joag-Dev and Proschan  pointed out that negatively associated (NA, in short) random variables are ND. They also presented an example in which possesses ND but does not possess NA. The another example which is ND but is not NA was provided by Wu  as follows.
Example 1. Let be a binary random variable such that for . Let take the values , , , and , each with probability . It can be verified that all the ND conditions hold. However, Hence, , and are not NA.
So we can see that ND is weaker than NA. A number of well-known multivariate distributions have the ND properties, such as multinomial, convolution of unlike multinomials, multivariate hypergeometric, dirichlet, dirichlet compound multinomial, and multinomials having certain covariance matrices. Because of the wide applications of ND random variables, the limiting behaviors of ND random variables have received more and more attention recently. A number of useful results for ND random variables have been established by many authors. We refer to Volodin  for the Kolmogorov exponential inequality, Asadian et al.  for Rosenthal's type inequality, Kim  for Hájek-Rényi type inequality, Amini et al. [20, 21], Ko and Kim , Klesov et al. , and Wang et al.  for almost sure convergence, Amini and Bozorgnia , Kuczmaszewska , Taylor et al. , Zarei and Jabbari , Wu [16, 29], Sung , and Wang et al.  for complete convergence, Wang et al.  for exponential inequalities and inverse moment, Shen  for strong limit theorems for arrays of rowwise ND random variables, Shen  for strong convergence rate for weighted sums of arrays of rowwise ND random variables, and so on. 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 .
In this section, we will present some important lemmas which will be used to prove the main results of the paper.
Lemma 2 (cf. Bozorgnia et al. ). Let the random variables be ND, and let be all nondecreasing (or all nonincreasing) functions. Then random variables are ND.
Lemma 3 (cf. Asadian et al. ). Let , and let be a sequence of ND random variables with and for every . Then there exists a positive constant depending only on such that for every ,
The following concept of stochastic domination will be used in this work.
Definition 4. 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 .
By the definition of stochastic domination and integration by parts, we can get the following property for stochastic domination. For the details of the proof, one can refer to Wu [36, 37] or Shen and Wu .
Lemma 5. 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.
3. Main Results and Their Proofs
Unless otherwise specified, we assume throughout the paper that is defined by (2). For any function , we use to denote all continuity points of the function on . The norm is the Eucledean 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 strong consistency of the fixed design regression estimator .
Theorem 6. Let be a sequence of mean zero ND random variables, which is stochastically dominated by a random variable . Assume that conditions ()–() hold true. If for some , and if there exist some such that then for any ,
Proof. For and , we have by (1) and (2) that
Since , hence for any , there exist a and such that . If we take in (10), we can get that
Therefore, we have by conditions ()–() that
For a fixed design point , without loss of generality, we assume that (otherwise, we use and instead of , resp., and note that ).
By (12), we can see that in order to prove (9), we only need to show that where .
For fixed , choose and some positive integer (to be specified later). Denote, for , It is easy to check that . Hence, to prove (13), it suffices to show that, for any ,
By and , we can see that Hence, to prove ., it suffices to show that It is easily seen that, for fixed and , are still ND random variables by Lemma 2. Applying Lemma 3, we have for that Taking , we can get that
For , we have by 's inequality, Lemma 5, and condition that
For , if , we have by the proof of (20) that if , we have by that By (18)–(22), we can get (17), which together with (16) imply that .
Next, we will prove that . Since , implies that there are at least ’s such that . Hence, which is summable if we choose large integer such that . Hence, . by Borel-Cantelli lemma.
Since , implies that there are at least ’s such that . Similarly, we have . as .
Finally, we will prove that . as . It is easily seen that To prove that . as , it suffices to show that . and . as . Firstly, we will show that Denote that . For , it follows by Lemma 5 and that which implies that is a Cauchy sequence in , and hence there exists a random variable such that and . It follows by Lemma 5 and again that which together imply that ., and thus (25) holds. Similarly, we can get (26). Therefore, . and . follow by (25)-(26) and the Kronecker's lemma immediately. This completes the proof of the theorem.
As an application of Theorem 6, we give the strong consistency for the nearest neighbor estimator of . Without loss of generality, putting , taking , , and . For any , we rewrite as follows: If , then is permuted before when .
Let , the nearest neighbor weight function estimator of in model (1) is defined as follows: where
Based on the notations above, we can get the following result by using Theorem 6.
Corollary 7. Let be a sequence of mean zero ND random variables, which is stochastically dominated by a random variable . Assume that is continuous on the compact set . If for some and if there exists some such that , then for any ,
Proof. It suffices to show that the conditions of Theorem 6 are satisfied. Since is continuous on the compact set , hence is uniformly continuous on the compact set , which implies that is bounded on the set .
For any , if follows from the definition of and that Hence, conditions ()–() and (8) are satisfied. By Theorem 6, we can get (32) immediately. This completes the proof of the corollary.
The authors are most grateful to the Editor Michiel Bertsch and anonymous referee for careful reading of the paper and valuable suggestions which helped in improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (11201001, 11171001), the Natural Science Foundation of Anhui Province (1208085QA03, 1308085QA03), Applied Teaching Model Curriculum of Anhui University (XJYYXKC04), Doctoral Research Start-up Funds Projects of Anhui University, Students Innovative Training Project of Anhui University (201310357004), and the Students Science Research Training Program of Anhui University (KYXL2012007, kyxl2013003).
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