Research Article | Open Access

Yu Zhang, Xinsheng Liu, Mohamed Sief, "Strong Consistency of Estimators in a Partially Linear Model with Asymptotically Almost Negatively Associated Errors", *Discrete Dynamics in Nature and Society*, vol. 2020, Article ID 2934914, 19 pages, 2020. https://doi.org/10.1155/2020/2934914

# Strong Consistency of Estimators in a Partially Linear Model with Asymptotically Almost Negatively Associated Errors

**Academic Editor:**Stefan Balint

#### Abstract

This paper studies a heteroscedastic partially linear regression model in which the errors are asymptotically almost negatively associated (AANA, in short) random variables with not necessarily identical distribution and zero mean. Under some mild conditions, we establish the strong consistency of least squares estimators, weighted least squares estimators, and the ultimate weighted least squares estimators for the unknown parameter, respectively. In addition, the strong consistency of the estimator for nonparametric component is also investigated. The results derived in the paper include the corresponding ones of independent random errors and some dependent random errors as special cases. At last, two simulations are carried out to study the numerical performance of the strong consistency for least squares estimators and weighted least squares estimators of the unknown parametric and nonparametric components in the model.

#### 1. Introduction

Consider the following heteroscedastic partially linear regression model:where , are known and nonrandom design points, is an unknown parameter, and are unknown functions defined on a compact set , and are random errors.

Model (1) belongs to a kind of model called partially linear model which was introduced by Engle et al. [1] to analyse the relationship between temperature and electricity usage. Since then, many statisticians pay attention to studying partially linear regression models. Under the case of independent random errors, Hu et al. [2] studied the asymptotic normality of DHD estimators in a partially linear model; Hu [3] established the strong consistency and mean consistency of the estimators for and in model (1) with ; Gao et al. [4] established the asymptotic normality for the least squares estimators and weighted least squares estimators of based on the family of nonparametric estimators for and in model (1); Chen et al. [5] investigated the strong consistency of the estimators in model (1); and so on. Under the case of dependent random errors, Zeng and Liu [6] studied the asymptotic properties of the estimators for parametric and nonparametric parts in a partially linear model with NSD errors; Wang et al. [7] established the mean consistency, complete consistency, and uniform complete consistency of the estimators in a partially linear model based on -mixing errors. Wang et al. [8] and Wu and Wang [9] discussed the moment consistency and strong consistency for least squares estimators and weighted least squares estimators of and in a partially linear model with -mixing errors. Pan et al. [10] obtained the mean consistency and complete consistency of the estimators for and in model (1) with under mixingale errors; Hu [11] obtained the mean consistency and complete consistency of the estimators for and in model (1) with under linear time series errors; Liang and Jing [12] studied the asymptotic normality of the least squares estimators and the weighted least squares estimators in model (1) with martingale difference errors and linear process errors; Baek and Liang [13] investigated the strong consistency and asymptotic normality of the estimators in model (1) under negatively associated samples; Zhou et al. [14] derived the moment consistency of the estimators in model (1) with negatively associated errors; and so forth. For more studies on the asymptotic properties of the estimators in regression models, one can refer to [15, 16]. Asymptotically almost negatively associated sequences are widely used dependent sequences which include independent and negatively associated sequences as special cases. So, to study the limit properties of asymptotically almost negatively associated sequences has more theoretical significance and application value. The concept of asymptotically almost negatively associated sequences of random variables was introduced by Chandra and Ghosal [17] as follows.

*Definition 1. *A sequence of random variables is called asymptotically almost negatively associated (AANA, in short) if there exists a non-negative sequence as such thatfor all and for all coordinate-wise nondecreasing continuous functions and whenever the variances exist.

Chandra and Ghasal [17] pointed out that the family of AANA sequences contains negatively associated (NA, in short, see [18]) sequences (with ) and some more sequences of random variables which are not much deviated from being negatively associated. Two examples of AANA sequences which are not NA were constructed by Chandra and Ghosal: (see Chandra and Ghosal [17]) and (see [19]), where *Î·*_{1}, *Î·*_{2}, â€¦ are independent and identically distributed random variables, , and as .

Many applications of AANA sequences have been found. Chandra and Ghosal [17] derived the Kolmogorov-type inequality and Marcinkiewcz-Zygmund type strong laws of large numbers. An [20] studied the complete moment convergence of weighted sums for processes under AANA assumptions. Wang et al. [21] investigated the large deviation and Marcinkiewicz type strong law of large numbers for AANA sequences. Ko et al. [22] established the HÃ¡jeck-RÃ©nyi inequalities for AANA sequences. Shen and Wu [23] investigated the strong law of large numbers for AANA sequences. Yuan and An [24] provided some Rosenthal type inequalities for maximum partial sums of AANA sequences. Wang et al. [25] studied the complete convergence for weighted sums of arrays of rowwise AANA random variables. Xi et al. [26] investigated the *L** ^{P}* convergence and complete convergence for weighted sums of AANA random variables. Chen et al. [27] obtained the strong laws of large numbers for the weighted sums of AANA sequences; Hu and Zhang [28] obtained the strong consistency of M-estimator in the linear regression model with AANA errors; Zhang et al. [29] established the weak consistency of M-estimator in the linear regression model with AANA errors; and so on.

However, we have not found the studies on the strong consistency of the estimators for parametric and nonparametric components in model (1) with AANA random errors in the literature. In this paper, we will consider the estimation problem for model (1) under the assumption that the errors are AANA sequences of random variables with not necessarily identical distribution and zero mean. The strong consistency of least squares estimators, weighted least squares estimators, and the ultimate weighted least squares estimators for is derived, respectively, based on some mild conditions. In addition, the strong consistency of the estimators for and is also studied, respectively. These results extend and improve the corresponding ones for independent and identically distributed random errors and some dependent random errors.

The following concept of stochastic domination will be used in this work.

*Definition 2. *A sequence of random variables is said to be stochastically dominated by a random variable if there exists a positive constant such thatfor all and .

The remainder of this paper is organized as follows. The least squares estimators, weight least squares estimators, and ultimate weighted least squares estimators of based on the family of nonparametric estimators for and and some assumptions are introduced in Section 2. The main results are given in Section 3. We give some preliminary lemmas in Section 4. We provide the proofs of the main results in Section 5. Two simulations are presented in Section 6.

Throughout this paper, let , , and be positive constants whose values may vary at different places. stands for almost sure convergence, , and .

#### 2. Estimation and Assumptions

Assume that satisfy model (1) and is a measurable weight function on the compact set . Denote , , , and .

For model (1), by , one can get that for . Hence, for any given , we define an estimator of given by

To estimate , we seek to minimize

The minimum point of is found aswhere is called a least squares (LS, in short) estimator of . When the random errors are heteroscedastic, we modify to a weighted least squares (WLS, in short) estimator:

When , we have . Hence, the estimator of can be defined bywhere is a measurable weight function on . In general, one assumes that . Therefore, the ultimate weighted least squares estimators (UWLS, in short) of iswhere , , .

From (4)â€“(9), we further define

To obtain our results, the following assumptions are sufficient.â€‰(i).(ii).(iii) satisfies the first-order Lipschitz condition on compact subset .(iv) and are continuous functions on compact subset .(v).â€‰â€‰, .â€‰(i).(ii) for any .â€‰(i).(ii) for some.â€‰â€‰ satisfies the assumptions and replacing and by and , respectively.â€‰â€‰.

*Remark 1. *The assumptions are used in Chen et al. [5], Baek and Liang [13], Zhou et al. [30], and so forth. From (i) and (ii) of , it follows that

*Remark 2. *The following two weight functions satisfy the assumptions â€“:where , , , , is the Parzenâ€“Rosenblatt kernel function (see [31]), and is a bandwidth parameter.

#### 3. Statement of Main Results

In this section, let be an AANA sequence of random variables with zero mean and mixing coefficients , which is stochastically dominated by a random variable .

Theorem 1. *Suppose that (i), (ii), (iv), and (v) of , , , and hold. If , then*

*Remark 3. *Since independent sequences are special AANA sequences (see Chandra and Ghosal [17]), Theorem 1 extends and improves the corresponding results of Chen et al. [5] for identically distributed independent random errors to the case of not necessarily identically distributed AANA setting.

Theorem 2. *Suppose that the conditions of Theorem 1 are satisfied. Assume further that ; then,*

*Remark 4. *When , AANA sequences are reduced to NA sequences (see Chandra and Ghosal [17]). Therefore, Theorems 1 and 2 also hold for not necessarily identically distributed NA random errors.

Theorem 3. *Let . Suppose that , , andâ€“hold. If , then*

*Remark 5. *As independent sequences are special AANA sequences, Theorem 3 extends and improves the corresponding results of Chen et al. [5] for identically distributed independent random errors to the case of not necessarily identically distributed AANA random errors.

Theorem 4. *Suppose that the conditions of Theorem 3 are satisfied. Assume further that ; then,*

*Remark 6. *As NA sequences are special AANA sequences with , Theorems 3 and 4 also hold for not necessarily identically distributed NA random errors.

#### 4. Preliminary Lemmas

To prove the results of this paper, the following lemmas are needed.

Lemma 1 (see [24]). *If is an AANA sequence with mixing coefficients , then is still an AANA sequence with mixing coefficients , where are nondecreasing or nonincreasing functions.*

Lemma 2 (see [28]). *Let be an AANA sequence of random variables with zero mean and mixing coefficients . If , and Then,for any .*

Lemma 3. *Let be an AANA sequence of random variables with zero mean and mixing coefficients . Assume that for and ; then,for any .*

*Proof. *From Lemma 2 and , it follows thatTake ; then, . By (25), we derive thatSince , we haveThis completes the proof of Lemma 3.

Lemma 4 (see [21]). *Let be an AANA sequence of random variables with zero mean, mixing coefficients , and . If , then*

Lemma 5. *Let be an AANA sequence of random variables with mixing coefficients and . Denote for some . Suppose that**Then,*

*Proof. *Denote . By Lemma 1, we know that is still an AANA sequence of random variables with mixing coefficients . By (29) and (30), we haveNote thatHence,where .

By (30), we obtain thatHence,Thus, it follows from (29) and (31) thatBy Lemma 4, we derive thatHence, it follows from (33) thatBy (29), we obtain thatHence, it follows from Borelâ€“Cantelli lemma thatTherefore, (32) follows from (40) and (42).

This completes the proof of Lemma 5.

Lemma 6. *Let be an AANA sequence of random variables with mixing coefficients and , which is stochastically dominated by a random variable . For any , denote**If for some , then*

*Proof. *By Lemma 1, we know that and are still AANA random variables with mixing coefficients. Denoteand then . By Lemma 4, we know that and are still AANA sequences of random variables with mixing coefficients. Since is stochastically dominated by a random variable , for some fixed , we haveHence,Denote for and . Then,Since , we haveThus,DenoteThen,Hence, it follows from (50), (52), (53), and Lemma 5 thatSimilar to the proof of (54), we can get thatTherefore,This completes the proof of Lemma 6.

Lemma 7 (see [32]). *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. Thus,where is a positive constant.*

Lemma 8. *Let be an AANA sequence of random variables with mixing coefficients and , which is stochastically dominated by a random variable with for . Then,*

*Proof. *Denoteand then we obtain by thatHence, it follows by the Borelâ€“Cantelli lemma thatThus, to prove (59), it suffices to show thatBy Lemma 1, we know that is still an AANA sequence of random variables. Hence, by Lemma 4 and Kronecker lemma, to prove (63), we only need to show thatBy the Markov inequality, Lemma 4, and , we haveTherefore, (59) follows.

This completes the proof of Lemma 8.

#### 5. Proofs of the Main Results

By (1), (6), (7), and (9), we derive thatwhere , , and .

*Proof of Theorem 1. *We prove (16) first. By (67), we can get thatHence, to prove (16), it suffices to show thatFirstly, we proveIn view of (69), we haveFrom (ii) and (v) of and (13), it follows thatDenotefor any . Write , ; then, we know by Lemma 1 that and are still AANA sequences with mixing coefficients and zero mean. By (ii) and (iv) of and Lemma 7, we havefor .

Hence, for any and sufficiently large , by Lemma 4, we haveHence, it follows from the Borelâ€“Cantelli lemma thatBy Lemma 6, we obtain thatBy (73) and (78), we derive thatHence, combining (77), (79), and (80), we haveThus, (71) follows.

Secondly, we proveIn view of (69), we haveFrom (ii) of , , and (13), it follows that