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

)is 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. )e 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.


Introduction
Consider the following heteroscedastic partially linear regression model: where σ 2 i � f(u i ), (x i , t i , u i ) are known and nonrandom design points, β is an unknown parameter, f(·) and g(·) are unknown functions defined on a compact set A, and e i , 1 ≤ i ≤ n 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 β and g(·) in model (1) with σ 2 i � σ 2 ; Gao et al. [4] established the asymptotic normality for the least squares estimators and weighted least squares estimators of β based on the family of nonparametric estimators for g(·) and f(·)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 φ-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 β and f(·) in a partially linear model with ρ-mixing errors. Pan et al. [10] obtained the mean consistency and complete consistency of the estimators for β and g(·) in model (1) with σ 2 i � σ 2 under L q mixingale errors; Hu [11] obtained the mean consistency and complete consistency of the estimators for β and g(·) in model (1) with σ 2 i � σ 2 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. e concept of asymptotically almost negatively associated sequences of random variables was introduced by Chandra and Ghosal [17] as follows.

Definition 1.
A sequence X n , n ≥ 1 of random variables is called asymptotically almost negatively associated (AANA, in short) if there exists a non-negative sequence q(n) ⟶ 0 as n ⟶ ∞ such that Cov f X n , g X n+1 , X n+2 , . . . , X n+k ≤ q(n) Var f X n Var g X n+1 , X n+2 , . . . , X n+k for all n, k ≥ 1and for all coordinate-wise nondecreasing continuous functions f and g 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 q(n) � 0, n ≥ 1) 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: ξ n � (1 + a 2 n ) − 1/2 (η n + a n η n+1 ) (see Chandra and Ghosal [17]) and ζ n � η n + a n η n+1 (see [19]), where η 1 , η 2 , . . . are independent and identically distributed N(0, 1) random variables, a n > 0, and a n ⟶ 0 as n ⟶ ∞.
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 Hájeck-Rényi 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. e strong consistency of least squares estimators, weighted least squares estimators, and the ultimate weighted least squares estimators for β is derived, respectively, based on some mild conditions. In addition, the strong consistency of the estimators for g(·) and f(·) is also studied, respectively. ese results extend and improve the corresponding ones for independent and identically distributed random errors and some dependent random errors. e following concept of stochastic domination will be used in this work.

Definition 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. e remainder of this paper is organized as follows. e least squares estimators, weight least squares estimators, and ultimate weighted least squares estimators of β based on the family of nonparametric estimators for g(·) and f(·) and some assumptions are introduced in Section 2. e 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. roughout this paper, let C, C 1 , and C 2 be positive constants whose values may vary at different places. ⟶ a.s. stands for almost sure convergence, f + � max 0, f , and f − � min 0, f .

Estimation and Assumptions
Assume that Y i , x i , t i ∈ A, u i ∈ A, 1 ≤ i ≤ n satisfy model (1) and W ni (t) � W ni (t; t 1 , t 2 , . . . , t n ) is a measurable weight function on the compact set A. Denote (1), by Ee i � 0, one can get that Hence, for any given β, we define an estimator of g(·) given by To estimate β, we seek to minimize 2 Discrete Dynamics in Nature and Society e minimum point of Q(β) is found as where β LS is called a least squares (LS, in short) estimator of β. When the random errors are heteroscedastic, we modify β LS to a weighted least squares (WLS, in short) estimator: When Hence, the estimator of f(·) can be defined by where W ni (u) � W ni (u; u 1 , u 2 , . . . , u n ) is a measurable weight function on A. In general, one assumes that min 1≤i≤n |f n (u i )| > 0. erefore, the ultimate weighted least squares estimators (UWLS, in short) of β is where U 2 n � n i�1 c ni x 2 i , c ni � 1/f n (u i ), 1 ≤ i ≤ n. From (4)-(9), we further define To obtain our results, the following assumptions are sufficient.
satisfies the first-order Lipschitz condition on compact subset A. (iv) f(·) and g(·) are continuous functions on compact subset A. (1) for any δ > 0.
W ni (·) satisfies the assumptions (A 2 ) and (A 3 ) replacing t i and W ni by u i and W ni , respectively.

Remark 1.
e assumptions (A 1 ) are used in Chen et al. [5], Baek and Liang [13], Zhou et al. [30], and so forth. From (i) and (ii) of (A 1 ), it follows that Remark 2. e following two weight functions satisfy the assumptions (A 2 )-(A 4 ): where is the Parzen-Rosenblatt kernel function (see [31]), and H n is a bandwidth parameter.

Statement of Main Results
In this section, let e i , i ≥ 1 be an AANA sequence of random variables with zero mean and mixing coefficients q(i), i ≥ 1 , which is stochastically dominated by a random variable e.
Remark 3. Since independent sequences are special AANA sequences (see Chandra and Ghosal [17]), eorem 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.
Remark 5. As independent sequences are special AANA sequences, eorem 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 eorem 3 are satisfied. Assume further that
Remark 6. As NA sequences are special AANA sequences with q(i) ≡ 0, eorems 3 and 4 also hold for not necessarily identically distributed NA random errors.

Preliminary Lemmas
To prove the results of this paper, the following lemmas are needed.
Lemma 2 (see [28]). Let X i , i ≥ 1 be an AANA sequence of random variables with zero mean and mixing coefficients for any ε > 0.

Lemma 3. Let X i , i ≥ 1 be an AANA sequence of random variables with zero mean and mixing coefficients
for any ε > 0.
□ Lemma 4 (see [21]). Let X i , i ≥ 1 be an AANA sequence of random variables with zero mean, mixing coefficients . By Lemma 1, we know that X c i , i ≥ 1 is still an AANA sequence of random variables with mixing coefficients q(i), i ≥ 1 . By (29) and (30), we have Note that Hence, Hence, us, it follows from (29) and (31) that By Lemma 4, we derive that Hence, it follows from (33) that By (29), we obtain that Hence, it follows from Borel-Cantelli lemma that erefore, (32) follows from (40) and (42). is completes the proof of Lemma 5. □ Lemma 6. Let X i , i ≥ 1 be an AANA sequence of random variables with mixing coefficients q(i), i ≥ 1 and ∞ i�1 q 2 (i) < ∞, which is stochastically dominated by a random variable X. For any ε > 0, denote Discrete Dynamics in Nature and Society 5 Proof. By Lemma 1, we know that X i ′ , i ≥ 1 and X i ″ , i ≥ 1 are still AANA random variables with mixing coefficients q(i), i ≥ 1 . Denote and then |X i Hence, Since Denote en, 6 Discrete Dynamics in Nature and Society Hence, it follows from (50), (52), (53), and Lemma 5 that Similar to the proof of (54), we can get that is completes the proof of Lemma 6. □ Lemma 7 (see [32]). Let X n , n ≥ 1 be a sequence of random variables which is stochastically dominated by a random variable X. For any a > 0 and β > 0, the following two statements hold: where C 1 and C 2 are positive constants. us, where C is a positive constant.
Lemma 8. Let X i , i ≥ 1 be an AANA sequence of random variables with mixing coefficients q(i), i ≥ 1 and ∞ i�1 q 2 (i) < ∞, which is stochastically dominated by a random variable X with E|X| p < ∞ for 0 < p < 2. en, Proof. Denote X n ′ � − n (1/p) I X n ≤ − n (1/p) + X n I X n < n (1/p) + n (1/p) I X n ≥ n (1/p) , and then we obtain by Hence, it follows by the Borel-Cantelli lemma that X n � X n ′ a.s.
us, to prove (59), it suffices to show that By Lemma 1, we know that X n ′ − EX n ′ , n ≥ 1 is still an AANA sequence of random variables. Hence, by Lemma 4 and Kronecker lemma, to prove (63), we only need to show that Discrete Dynamics in Nature and Society By the Markov inequality, Lemma 4, and E|X| p < ∞, we have erefore, (59) follows. is completes the proof of Lemma 8.
Hence, for any μ > 0 and sufficiently large n, by Lemma 4, we have Hence, it follows from the Borel-Cantelli lemma that In view of (69), we have Discrete Dynamics in Nature and Society From (ii) of (A 1 ), (A 2 ), and (13), it follows that Hence, similar to the proof (71), one can derive (82). Finally, we prove In view of (69), we have By (iv) of (A 1 ), (A 3 ), and max 1≤i≤n as n ⟶ ∞.
is completes the proof of eorem 1.
is completes the proof of eorem 2.

□
Proof of eorem 3. Firstly, we prove (19). In view of (66), we have for any ε > 0. Write for 1 ≤ i ≤ n, and nj , 1 ≤ j ≤ n , and e (2) nj , 1 ≤ j ≤ n are still AANA sequences with mixing coefficients q(i), i ≥ 1 and zero mean. Note that By Lemma 7, we obtain that Hence, similar to the proof of (78), we can easily derive Hence, similar to the proof of (71), we can easily obtain that erefore, (19) follows from (96), (98), (106), and (107). Secondly, we prove (20). In view of (8), we have Next, we will prove Since We know by Lemma 1 that (e + i ) 2 and (e − i ) 2 are still AANA sequences with mixing coefficients q(i), i ≥ 1 . Observe that and η i is still an AANA sequence with mixing coefficients q(i), i ≥ 1 . Similar to the proof of (71), one can derive that By (iv) of (A 1 ) and (A 5 ), we have erefore, (109) follows. Note that Similar to the proof of (71), one can easily obtain that

Numerical Simulations
In this section, we will investigate the numerical performance of the strong consistency for the least squares estimators β LS and g n (t) and weighted least squares estimators β WLS and g n (t) with AANA random errors by two simulated examples. Two AANA sequences are given as follows: where η 1 , η 2 , · · · are independent and identically distributed N(0, 1) random variables and a i � (1/i 2 ). e two sequences have been proved to be AANA sequences but not NA sequences (see [17,19]). Next, we will apply the two sequences to the following two simulation examples, respectively.
6.1. Simulated Example 1. We will simulate a heteroscedastic partially linear model and the random errors are given by Sequence 1.
In particular, we take the weight function W ni (·) as the following nearest neighbor weight function (see [11]). Without loss of generality, let A � [0, 1] and t i � (i/n) (1 ≤ i ≤ n). For each t ∈ A, we rewrite as follows: Take k n � [n 0.4 ] and define the nearest neighbor weight function e sample sizes are taken as n � 200, 500, 1000, 1500, 2000, and 2500, respectively, and each case is repeated for 1000 times and the average values of β LS , g n (t), β WLS , and g n (t)are calculated as the estimators, respectively. en, we calculate the corresponding absolute errors |β LS − β|, |g n (t) − g(t)|, |β WLS − β|, and |g n (t) − g(t)| and the root mean square errors (RMSEs) of β LS , g n (t), β WLS , and g n (t), respectively. e above results are presented in Tables 1 and  2, and the curves of g(t), g n (t), and g n (t) are provided in Figures 1-2 where β � 2.5, g(t) � cos(πt), σ i � (f(u i )) 1/2 � (1 − 0.5 sin(πu i )) 1/2 , u i � x i � (− 1) i · i/n, 1 ≤ i ≤ n, and the random errors are given by Sequence 2.     g n (t) Figure 2: Curves of g(t) � sin(2πt) and g n (t) with β � 3.5 and n � 1500. 16 Discrete Dynamics in Nature and Society Table 3: e LS estimators of β and g(t) and the absolute errors and RMSEs of β LS and g n (t) with β � 2.5 and g(t) � cos(πt).    By the same estimating methods as model (6), we obtain the estimators of β and g n (t) in model (140) under different sample size n, |g n (t) − g(t)|, |β WLS − β|, and |g n (t) − g(t)| and the RMSE of β LS , g n (t), β WLS , and g n (t), respectively. e above results are provided in Tables 3 and 4, and the curves of g(t), g n (t), and g n (t) are presented in Figures 3  and 4, respectively.
It can be seen from Tables 1-4 that regardless of the values of t, the absolute errors of both the least squares estimators and weighted least squares estimators decrease gradually as the sample size n has good effects. e simulation results show the strong consistency of least squares estimators β LS and g n (t) and weighted least squares estimators β WLS and g n (t) in model (1) with AANA random errors. e simulation results also show that the absolute error of weighted least squares estimator is smaller than that of least squares estimator. Moreover, consistency is the basic standard that all estimators should meet, and it is the necessary condition to measure whether an estimator is feasible. AANA sequences are widely used dependent sequences which include independent and NA sequences as special cases. erefore, to study the consistency of estimators in regression models with AANA errors is of considerable significance.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.