The Consistency of Estimators in a Heteroscedastic Partially Linear Model with ρ − -Mixing Errors

: This paper studies a heteroscedastic partially linear model based on ρ − -mixing random errors, stochastically dominated and with zero mean. Under some suitable conditions, the strong consistency and p -th ( p > 0 ) mean consistency of least squares (LS) estimators and weighted least squares (WLS) estimators for the unknown parameter are investigated, and the strong consistency and p -th ( p > 0 ) mean consistency of the estimators for the non-parametric component are also studied. These results include the corresponding ones of independent, negatively associated (NA), and ρ ∗ -mixing random errors as special cases. At last, two simulations are presented to support the theoretical results.


Introduction
Consider the following heteroscedastic partially linear model: where σ 2 in = f (u in ), z in ∈ R, x in ∈ R p , u in ∈ R p , and (x in , z in , u in ) are known and nonrandom design points, β represents an unknown parameter, f (·) and h(·) represent unknown functions, which are defined on a compact set M ⊂ R p , y (t) (x in , z in ) stands for the t-th variables that can be observable at points (x in , z in ), and ε (t) (x in ), 1 ≤ t ≤ r, 1 ≤ i ≤ n stands for random errors.
In order to analyze the effect of temperature on electricity usage, Engle et al. [1] proposed the partially linear model Since then, many statisticians have studied partially linear regression models. The model (2) was further investigated by Heckman [2], Speckman [3], Gao [4], Härdle et al. [5], Hu et al. [6], Zeng and Liu [7], and so forth. Some applications of the model were given. Inspired by the model (2), a more general model was proposed by Gao et al. [8]: Gao et al. [8] established the asymptotic normality of least squares (LS) and weighted least squares (WLS) estimators for β based on the family of non-parametric estimators for h(·) and f (·) in the model (3). Baek and Liang [9] investigated the asymptotic property in the model (3) for negatively associated where ρ(T, U) = sup Var(X)Var(Y) : X ∈ L 2 (σ(T)), Y ∈ L 2 (σ(U)) , σ(T) and σ(U) are σ-fields that are generated by X j , j ∈ T and {X k , k ∈ U} respectively, L 2 (σ(T)) is the space of all square integral and σ(T)-measurable random variables, and L 2 (σ(U)) is defined in the same way.
On one hand, NA sequences have been widely applied to reliability theorem and multivariate statistical analysis (see [21,22]). On the other hand, some Markov Chains and moving average processes are ρ * -mixing sequences (see [23]). The concept of ρ * -mixing sequences is important in a lot of areas, for instance, finance, economics, and other sciences (see [24]). Therefore, studying ρ − -mixing sequences is of considerable significance.
Since Zhang and Wang [19] proposed the concept of ρ − -mixing sequences, many results on ρ − -mixing sequences have been established. One can refer to Zhang and Wang [19], Wang and Lu [25], and Yuan and An [26] for some moment inequalities and some limiting behavior; Zhang [20] and Zhang [27] for some central limit theorems; Chen et al. [28] for complete convergence for weighted sums of ρ − -mixing sequences; Zhang [29] for the complete moment convergence for the partial sum of ρ − -mixing moving average processes; Wu and Jiang [30] for almost sure convergence of ρ − -mixing sequences; and Xu and Wu [31] for an almost sure central limit theorem for the self-normalized partial sums.
However, we have not found studies on the model (1) under ρ − -mixing random errors in the literature. In the present paper, we will study the estimation problem for the model (1) based on the assumption that the errors are ρ − -mixing sequences that are stochastically dominated and zero mean. The strong consistency and mean consistency of LS estimators and WLS estimators for β and h(·) are established respectively based on some suitable conditions. The results obtained in the paper deal with independent errors as well as dependent errors as special cases.
Next, we will recall the definition of stochastic domination.
for some c > 0, every y ≥ 0 and each n ≥ 1.
The remainder of this paper is organized as follows. The LS estimators and WLS estimators of β based on the family of non-parametric estimators for h(·) and some conditions are introduced in Section 2. We give the main results in Section 3. Several lemmas are given in Section 4. We provide the proofs of the main results in Section 5. Two simulations are carried out in Section 6. We conclude the paper in Section 7. Throughout the paper, let C denote positive constants whose values may be different in various places. "i.i.d." stands for independent and identically distributed. · stands for the Euclidean norm.

Estimation and Conditions
Assume that y (t) (x in , z in ), z in ∈ R, x in ∈ M, u in ∈ M, 1 ≤ t ≤ r, 1 ≤ i ≤ n satisfies the model (1) and W nj (x) = W nj (x; x 1 , x 2 , · · · , x n ) is a weight function that is measurable on the compact set M. For simplicity and convenience, the model (1) can be written as For the model (5), one can get from E ε Thus, for any given β, we can define the non-parametric estimator of h(·) in terms of Hence, the LS estimators of β can be defined bŷ By (7), we haveβ When the random errors are heteroscedastic, we modifyβ r,n to a WLS estimator. We can define the WLS estimators of β in terms of By (9), we derive thatβ Taking into accountβ and In order to obtain the relevant theorems, several important conditions are given below.
(iii) f (·) and h(·) are continuous functions on compact set M.

Remark 1.
Conditions (C 1 )(i) (ii) are some regular conditions that are often imposed in studies of LS and WLS estimators in heteroscedastic partially linear models. One can refer to [5,8,9] and so on. (C 1 ) (iii) is mild and holds for most commonly used functions, such as polynomial and trigonometric functions (see [9]). Conditions (C 2 )-(C 4 ) are often applied to investigate strong consistency (see [9,33]) and mean consistency (see [10,16]). (C 2 )(ii) is weaker than the corresponding conditions of [16] and [33]. Thus, the above conditions are very mild. Moreover, by (C 1 )(i) (ii), one can get that and

Main Results
In this paper, let ε (t) i , 1 ≤ t ≤ r, 1 ≤ i ≤ n be a ρ − -mixing sequence with zero mean, which is stochastically dominated by a random variable ε.

Theorem 2.
Under the conditions of Theorem 1, in addition, if (C 4 ) holds, then as min(r, n) → ∞ and as min(r, n) → ∞ .
and lim Theorem 4. Under the conditions of Theorem 3, in addition, if (C 4 ) holds, then and lim Remark 2. Since ρ − -mixing sequences include NA (in particular, independent) and ρ * -mixing sequences, Theorems 1-4 also apply for NA and ρ * -mixing sequences.

Some Lemmas
From the definition of ρ − -mixing sequences, we can get the first lemma.
is a random sequence that is stochastically dominated by a random variable X, for every a > 0 and β > 0, we have , Therefore, where c is a positive constant.
and ε 2i . Without loss of generality, one can suppose that c ni (v) > 0. Hence, we know by Lemma 1 that The proof of (24) is similar to that of the Lemma 3.3 in Zhou and Hu [34]. By Lemma 3, we have Hence, for every s > p > 2, from the Markov inequality, Lemma 2 and E|ε| p < ∞, we get that Hence, it follows from (24) through (26) By the Borel-Cantenlli lemma, we obtain for any v ∈ M that as min(r, n) → ∞ . Therefore, (23) follows.

Proofs of the Main Results
By (5), (8), and (10), we derive that where e Proof of Theorem 1. We only need to prove (16) since the proof of (15) is analogous. By (32), we can get thatβ Observe that I (1) i . Hence, it follows from (C 1 )(i) and (ii) and (14) that max and Thus, by Lemma 4, we have as min(r, n) → ∞ . Note that Hence, it follows from (C 1 )(ii), (C 2 ), and (14) that and where α is the same as that in (C 2 )(ii). Thus, by Lemma 4, one can get that as min(r, n) → ∞ . By (14), we derive that By (C 1 )(iii), (C 2 )(i), and (C 3 ), we obtain that Thus, by (40)

Proof of Theorem 3.
We only need to prove (20) since the proof of (19) is analogous. By (33), we havê Hence, it follows by C p inequality that The rest of the proof is similar to the proof of (16), so we omitted the details here.
Proof of Theorem 4. We only need to prove (22) since the proof of (21) is analogous. By (43), we derive that r,n + J r,n .

Numerical Simulations
In this section, we will verify the validity of the theoretical results by two simulations.
In particular, we take the weight function W ni (·) as the following nearest neighbor weight function (see [11,35]). Without loss of generality, denote M = [0, 1] and For each x ∈ M, we rewrite |x 1 − x|, |x 2 − x|, · · · , |x n − x| as follows: Take k n = n 0.8 and define the nearest neighbor weight function as where the sample sizes are taken as n = 100, 600, 1200, 1900, 2700, and 3600 and the points x are taken as x = 0.2, 0.4, 0.6, and 0.8, respectively. We computeβ r,n − β andĥ r,n (x) − h(x) for 1000 times, respectively. The boxplots ofβ (LS) r,n − β are provided in Figures 1-4, the violin plots ofĥ r,n (x) − h(x) are provided in Figures 5-8, the curves of h(x) andĥ r,n (x) are provided in Figure 9, and the mean squared errors (MSE) ofβ

Conclusions
In this paper, we mainly investigated the asymptotic properties of the estimators for the unknown parameter and non-parametric component in the heteroscedastic partially linear model (1). A lot of authors have derived the asymptotic properties of the estimators in partially linear models   Table 4. The MSEs ofĥ r,n (x) with β = 3.5 and h(x) = cos(πx).

Conclusions
In this paper, we mainly investigated the asymptotic properties of the estimators for the unknown parameter and non-parametric component in the heteroscedastic partially linear model (1). A lot of authors have derived the asymptotic properties of the estimators in partially linear models with independent random errors (see [4][5][6]8,33]). However, in many applications, the random errors are not independent. Here, we assumed that the random errors are ρ − -mixing, which includes independent, NA, and ρ * -mixing random variables as special cases. Under some suitable conditions, the strong consistency and p-th (p > 0) mean consistency of the LS estimator and WLS estimator for the unknown parameter β were investigated, and the strong consistency and p-th (p > 0) mean consistency of the estimators for the non-parametric component h(·) were also studied. The results obtained in this paper include the corresponding ones of independent random errors, NA random errors (see [16]), and ρ * -mixing random errors as special cases. Furthermore, for the model (1), we carried out simulations to study the numerical performance of the asymptotic properties for the estimators of the unknown parameter and non-parametric component for the first time. ρ − -mixing sequences are widely used dependent sequences. Therefore, investigating the limit properties of the estimators in regression models under ρ − -mixing errors in future studies is an interesting subject.
Author Contributions: Methodology, Software, Writing-original draft, and Writing-review and editing, Y.Z.; Funding acquisition, Supervision, and Project administration, X.L.; Validation, Y.Z. and X.L. All authors have read and agreed to the published version of the manuscript.
Funding: This work was supported by the National Natural Science Foundation of China (61374183) and the Project of Guangxi Education Department (2017KY0720).

Conflicts of Interest:
The authors declare no conflict of interest.