ASYMTOTIC NORMALITY OF ESTIMATORS IN HETEROSCEDASTIC ERRORS-IN-VARIABLES MODEL FOR NA SAMPLES Ting Wang & Jing-jing Zhang

ASYMTOTIC NORMALITY OF ESTIMATORS IN HETEROSCEDASTIC ERRORS-IN-VARIABLES MODEL FOR NA SAMPLES

Ting Wang & Jing-jing Zhang^*

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, PR China.

∗Correspondence: [email protected] ABSTRACT

This article is concerned with the estimating problem of heteroscedastic partially linear errors-in-variables models.

We derive the asymptotic normality for estimators of the slope parameter and the nonparametric component in the case of known error variance with NA(negatively associated) random errors. Also, when the error variance is unknown, the asymptotic normality for the estimators of the slope parameter and the nonparametric component as well as variance function is considered under independent assumptions. Finite sample behavior of the estimators is investigated via simulations too.

𝐊𝐞𝐲𝐰𝐨𝐫𝐝𝐬: Partially linear errors-in-variables model, Negatively associated, Asymptotic normality, Heteroscedastic, Least-squares estimator.

𝑀𝑆𝐶: 62J12 ⋅ 62E20 1. INTRODUCTION

Consider the following heteroscedastic partially linear errors-in-variables (EV) model {𝑦𝑖= 𝜉𝑖𝛽 + 𝑔(𝑡𝑖) + 𝜀𝑖,

𝑥_𝑖= 𝜉_𝑖+ 𝜇_𝑖. (1)

where 𝜀𝑖= 𝜎𝑖𝑒𝑖, 𝜎𝑖2= 𝑓(𝑢𝑖), (𝜉𝑖, 𝑡𝑖, 𝑢𝑖) are nonrandom design points, (𝑡𝑖, 𝑥𝑖, 𝑦𝑖) are observed samples, 𝛽 is an unknown parameter to be estimated, {𝜉_𝑖} are the potential variables cannot be observed, {𝑦_𝑖} are the response variables, {𝑥𝑖} are observed with measurement errors {𝜇𝑖} and with 𝐸𝜇𝑖= 0, and {𝑒𝑖} are random errors and with 𝐸𝑒_𝑖= 0. Assume that there is a function ℎ(⋅) defined on closed interval [0,1] satisfying

𝜉𝑖= ℎ(𝑡𝑖) + 𝑣𝑖. (2)

where {𝑣_𝑖} are also unknown design points.

Model (1) and its special cases have been widely studied by many authors. When the {𝜉_𝑖} can be observed, 𝜎_𝑖²= 𝜎², and the errors {𝑒𝑖} are independent identically distribution(i.i.d), the model reduces to the homoscedastic partially linear regression model, which was studied by Engle et al (1986)[1]. When 𝑔(𝑡) ≡ 0, 𝜎_𝑖²= 𝑓(𝑢_𝑖), the model becomes into heteroscedastic partially linear regression model, which was extensively studied by Carroll (1982)[2], Robinson (1987)[3]. In addition, when 𝑔(𝑡) ≠ 0 , and the {𝜉𝑖} can not de directly observed, the model (1) degenerates into partially linear EV model, which can be seen in Cui and Li (1998)[4], Wang (1999)[5], Liang (1999)[6] and so on.

In recent decades, semi-parametric EV models have been widely concenred. Miao, Zhang and Wang(2013)[7] considered the strong consistency and asymptotic normality for the least square estimators in a linear EV regression model; Liu and Chen(2005)[8] discussed the consistency of estimators and derived the equivalence relation of weak or strong consistency for the estimators; Cui(2006)[9] summarized the T regression estimate and EM arithmetic in a linear EV regression model; Many of early results of the study of EV model can be seen in Fuller (1987)[10], Cheng and Van Ness (1999)[11] and Carrol (1995)[12].

In this paper, we consider the estimation problem for model (1) under the errors {𝑒_𝑖, 1} being mean zero negatively associated(NA) random variables. A finite family of random variables {𝑋𝑖, 1} is said to be NA random variables if for every pair of disjoint subsets A and B of {1,2,...,n}, we have

𝐶𝑜𝑣(𝑓1(𝑋𝑖, 𝑖 ∈ 𝐴), 𝑓2(𝑋𝑗, 𝑗 ∈ 𝐵))0

whenever 𝑓₁ and 𝑓₂ are coordinatewise increasing function and such that the covariance exists. An infinite family of random variables is NA if every finite subfamily is NA.

The NA view was introduced by Alam and Saxena (1981)[13], and Joag-Dev and Proschan (1983)[14]

discovered the the character of multivariate distribution of NA sequence and discovered fundamental properties; Liang (2000)[15] discovered complete convergence; NA sequence not only has been applied in the multivariat statistical analysis, but also in the oceans, weather, and other engineering fields, risk analysis and time series analysis just as the same as other positive and negative dependent sequence. However, there are few asymptotic results for the estimators of parametric and nonparametric components in partial linear EV model regressions under NA error’s structure.

The paper is organized as follows. In Section 2, we list some assumptions. The main results are given in

Section 3. A simulation study is presented in section 4. Some preliminary lemmas are stated in Section 5. Proofs of the main results are provided in Sections 6.

2. ASSUMPTIONS

First, we assume that {𝑡_𝑖, ℎ_𝑖, 𝑣_𝑖, 𝑔_𝑖, 𝜀_𝑖, 𝜇_𝑖, 𝜉_𝑖, 1} satisfy model (1), and that 𝑊_𝑛𝑖(⋅)(1) are some weight functions defined on 𝐼 and set ℎ̃_𝑖= ℎ(𝑡_𝑖) − ∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)ℎ(𝑡_𝑗) , 𝑣̃_𝑖= 𝑣_𝑖− ∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝑣_𝑗 , 𝑔̃_𝑖= 𝑔(𝑡_𝑖) −

∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝑔(𝑡_𝑗) , 𝜀̃_𝑖= 𝜀_𝑖− ∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝜀_𝑗 , 𝜇̃_𝑖= 𝜇_𝑖− ∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝜇_𝑗 and 𝜉̃_𝑖= 𝜉_𝑖− ∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝜉_𝑗. Then, we shall list some conditions, which will be used in the paper.

•

- Let {𝑒𝑖, 1} be a sequence of NA random variables with mean zero, and let {𝜇𝑖, 1} be a sequence of independent random variables with mean zero, and {𝑒_𝑖, 1} is independent with {𝜇_𝑖, 1}. Assume that 𝐸𝑒_𝑖²= 1, sup_𝑖𝐸|𝑒_𝑖|^𝑝< ∞, for some 𝑝 > 4, sup_𝑖𝐸|𝜇_𝑖|^𝑝< ∞, for some 𝑝 > 4, and the 𝐸𝜇_𝑖²= Ξ_𝜇²> 0 is known.

- Let both of {𝑒𝑖, 1} and {𝜇𝑖, 1} be sequences of independent random variables with mean zero, 𝐸𝑒_𝑖²= 1, 𝐸𝜇_𝑖²= Ξ_𝜇²> 0 and sup_𝑖𝐸𝑒_𝑖⁶+ sup_𝑖𝐸𝜇_𝑖⁶< ∞. {𝜇_𝑖, 1} is independent of {𝑒_𝑗, 1}.

• Let {𝑣𝑖, 1} in condition (1.2) be a sequence satisfying - lim_𝑛→∞𝑛⁻¹∑^𝑛_𝑖=1𝑣_𝑖²= Σ₀(0 < Σ₀< ∞);

- lim𝑛→∞sup𝑛(√𝑛log𝑛)⁻¹⋅ max1| ∑^𝑚_𝑖=1𝑣𝑗_𝑖| < ∞.

•

- 0 < 𝑚₀min₁𝑓(𝑢_𝑖)max₁𝑓(𝑢_𝑖)₀< ∞;

- 𝑓(⋅), 𝑔(⋅) and ℎ(⋅) are continuous function and satisfy the first-order Lipschitz condition on 𝐼.

• The probability weight functions 𝑊_𝑛𝑗(𝑡_𝑖) be weight functions defined on [0,1] and satisfy - max₁∑^𝑛_𝑖=1𝑊_𝑛𝑗(𝑡_𝑖) = 𝑂(1);

- max₁∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝐼(|𝑡_𝑖− 𝑡_𝑗| > 𝑛^−1/4) = 𝑜(𝑛^−1/4);

- max_1,𝑗𝑊_𝑛𝑗(𝑡_𝑖) = 𝑜(𝑛^−1/2log⁻¹𝑛), - max1,𝑗𝑊𝑛𝑗(𝑡𝑖) = 𝑂(𝑛^−𝑠).

• Let 𝑊̂_𝑛𝑖(⋅)(1) be weight functions defined on 𝐼. Conditons A3(i)(ii)(iv) are satisfied replacing 𝑡_𝑖 and 𝑊𝑛𝑖 by 𝑢𝑖 and 𝑊̂_𝑛𝑖, respectively.

Remark 2.1 Conditions (A0)-(A3) are standard regularity conditions and used commonly in the literature, see Gao et al.(1994)[16] and Chen et al.(1998)[17];

Remark 2.2 Under some mild conditions, the following two weight functions satisfy hypothesis (A3):

𝑊_𝑛𝑖⁽¹⁾(𝑡) =¹

ℎ∫_𝑠^𝑠𝑖

𝑖−1𝐾(^𝑡−𝑠

ℎ_𝑛)𝑑𝑠, 𝑊_𝑛𝑖⁽²⁾(𝑡) = 𝐾(^{𝑡−𝑡𝑖}

ℎ_𝑛)[∑^𝑛_𝑗=1𝐾(^{𝑡−𝑡𝑖}

ℎ_𝑛)]⁻¹.

where 𝑠𝑖= (𝑡𝑖+ 𝑡𝑖−1)/2, 𝑖 = 1,2, . . . , 𝑛 − 1, 𝑠0= 0, 𝑠𝑛= 1, 𝐾(⋅) is the Parzen-Rosenblatt kernel function, we can see Parzen(1962)[18], and the ℎ_𝑛 is a bandwidth parameter.

3. MAIN RESULTS

For model (1), we want to seek the estimator of 𝛽 and 𝑔(⋅). Firstly, when the error are homoscedastic and the 𝜉_𝑖 can

be observed, we can apply the least squares estimation method to estimate the parameter 𝛽. On the hand, we assume the parameter 𝛽 is known, and then to estimate 𝑔(⋅); for each given 𝛽, we have 𝑔(𝑡𝑖) = 𝐸(𝑦𝑖− 𝑥𝑖𝛽),1. Therefore, based on the (𝑥𝑖, 𝑡𝑖, 𝑦𝑖), we can define the estimator of 𝑔(⋅), that is 𝑔𝑛∗(𝑡, 𝛽) = ∑^𝑛_𝑖=1𝑊𝑛𝑖(𝑡)(𝑦𝑖− 𝑥𝑖𝛽). Then, based on the model (1), we can also define the LSE of 𝛽 by following formula:

∑^𝑛_𝑖=1[𝑦_𝑖− 𝑥_𝑖𝛽 − 𝑔_𝑛^∗(𝑡_𝑖, 𝛽)]²− Ξ_𝜇²𝛽²= 𝑚𝑖𝑛!

On the other hand, under this condition of partially linear EV model, Liang et al.(1999)[?] improved the LSE on the basis of the usually partially linear model, and employ the estimator of parameter 𝛽, write that

𝛽̂𝐿= [∑^𝑛_𝑖=1(𝑥̃_𝑖²− Ξ𝜇2)]⁻¹∑^𝑛_𝑖=1𝑥̃𝑖𝑦̃𝑖. (1)

where 𝑥̃_𝑖= 𝑥_𝑖− ∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝑥_𝑗, 𝑦̃_𝑖= 𝑦_𝑖− ∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝑦_𝑗.

Secondly, when the errors are heteroscedastic, we consider two different cases according to 𝑓(⋅). If 𝜎_𝑖²= 𝑓(𝑢𝑖) are known, then the 𝛽̂𝐿 is medified to be the weighted least-squares estimator (WLSE)

𝛽̂𝑊₁ = [∑^𝑛_𝑖=1𝜎_𝑖⁻²(𝑥̃_𝑖²− Ξ𝜇2)]⁻¹∑^𝑛_𝑖=1𝜎_𝑖⁻²𝑥̃𝑖𝑦̃𝑖. (2) In fact, the 𝜎_𝑖²= 𝑓(𝑢𝑖) are unknown and must be estimated. In the case, suppose that 𝐸𝑒_𝑖²= 1, we have 𝐸[𝑦𝑖− 𝜉𝑖𝛽 − 𝑔(𝑡𝑖)]²= 𝑓(𝑢𝑖). Therefore, the estimator of 𝑓(𝑢𝑖) can be defined by

𝑓̂𝑛(𝑢𝑖) = ∑^𝑛_𝑗=1𝑊̂_𝑛𝑗(𝑢𝑖)(𝑦̃𝑗− 𝑥̃𝑗𝛽̂𝐿)²− Ξ𝜇2𝛽̂𝐿2. (3) For convenience, we assume that min₁𝑓̂_𝑛(𝑢_𝑖) > 0. Then we can define a nonparametric estimator of 𝜎_𝑖², 𝜎̂_𝑛𝑖² = 𝑓̂_𝑛(𝑢_𝑖).

In consequence, when the errors are heteroscedastic and unknown, the WLSE of 𝛽 is

𝛽̂𝑊₂ = [∑^𝑛_𝑖=1𝜎̂_𝑛𝑖⁻²(𝑥̃_𝑖²− Ξ𝜇2)]⁻¹∑^𝑛_𝑖=1𝜎̂_𝑛𝑖⁻²𝑥̃𝑖𝑦̃𝑖. (4) Meanwhile, using 𝛽̂𝐿, 𝛽̂𝑊₁, 𝛽̂𝑊₂, we can define the three estimators for 𝑔(⋅):

𝑔̂𝐿(𝑡) = ∑^𝑛_𝑖=1𝑊𝑛𝑖(𝑡)(𝑦𝑖− 𝑥𝑖𝛽̂𝐿), (5)

𝑔̂𝑊₁(𝑡) = ∑^𝑛_𝑖=1𝑊𝑛𝑖(𝑡)(𝑦𝑖− 𝑥𝑖𝛽̂𝑊₁), (6)

𝑔̂𝑊₂(𝑡) = ∑^𝑛_𝑖=1𝑊𝑛𝑖(𝑡)(𝑦𝑖− 𝑥𝑖𝛽̂𝑊₂). (7)

In this paper, we provide some notions and a definition that will be used in the process of proof.

𝜂𝑖= 𝜀𝑖− 𝜇𝑖𝛽, 𝑆𝑛2= ∑^𝑛_𝑖=1𝜉̃_𝑖², 𝑇𝑛2= ∑^𝑛_𝑖=1𝜎_𝑖⁻²𝜉̃_𝑖²,

𝑆_1𝑛² = ∑^𝑛_𝑖=1(𝑥̃_𝑖²− Ξ𝜇2), Σ_1𝑛² = V𝑎𝑟[∑^𝑛_𝑖=1𝜎_𝑖⁻²(𝜉̃𝑖+ 𝜇𝑖)(𝜀𝑖− 𝜇𝑖𝛽)],

Γ_𝑛²(𝑡) = V𝑎𝑟[∑^𝑛_𝑖=1𝑊_𝑛𝑖(𝑡)(𝜀_𝑖− 𝜇_𝑖𝛽)], Δ²_𝑛(𝑢) = ∑^𝑛_𝑖=1𝑊̂_𝑛𝑖²(𝑢)V𝑎𝑟[(𝜀_𝑖− 𝜇_𝑖𝛽)²]. (8)

Definition 3.1 Let {𝑋_𝑡, 𝑡 = 0, ±1, ±2, ⋯ } be a strictly stationary time series. For 𝑛 = 1,2, ⋯, define

𝜌(𝑛) = sup

𝑋∈𝐿²(𝐹_−∞⁰ ),𝑌∈𝐿²(𝐹_𝑛^∞)

|𝐶𝑜𝑟𝑟(𝑋, 𝑌)|

where 𝐹_𝑖^𝑗 denotes the 𝜎 -algebra generated by {𝑋𝑡, 𝑖}, and 𝐿²(𝐹_𝑖^𝑗) consists of 𝐹_𝑖^𝑗-measurable random variables with finite second moment.

When 𝑓(⋅) is known, we give tne asymptotic normality for least-sqares estimators and weighted least- squares estimators of 𝛽 and 𝑔(⋅).

Theorem 3.1 Suppose that (A0)(i), (A1), (A2) and (A3) are satisfied. Then we have • If Σ𝑛2, then 𝑆𝑛2(𝛽̂𝐿− 𝛽)/Σ𝑛→^𝐷 𝑁(0,1);

• If Σ_1𝑛² , then 𝑇_𝑛²(𝛽̂_𝑊1− 𝛽)/Σ_1𝑛→^𝐷 𝑁(0,1).

Theorem 3.2 Suppose that (A0)(i), (A1), (A2) and (A3) are satisfied. If 𝑛𝛤_𝑛²(𝑡) → ∞ and ∑^𝑛_𝑖=1𝑊_𝑛𝑖²(𝑡) = 𝑂(𝛤_𝑛²(𝑡)), then we have

• [𝑔̂_𝐿(𝑡) − 𝐸𝑔̂_𝐿(𝑡)]/Γ_𝑛(𝑡) →^𝐷 𝑁(0,1);

• [𝑔̂𝑊₁(𝑡) − 𝐸𝑔̂𝑊₁(𝑡)]/Γ𝑛(𝑡) →^𝐷 𝑁(0,1).

Remark 3.1 According to Zhang and Liang(2013) Remark (3.1), we think ∑²_𝑛𝐶𝑛, ∑_1𝑛𝐶𝑛𝑎𝑛𝑑𝑛𝛤_𝑛²(𝑡) → ∞ is reasonable.

When 𝑓(⋅) is unknown, we give tne asymptotic normality for the estimators of 𝛽, 𝑔(⋅) and 𝑓(⋅) under the {𝑒_𝑖, 1} is an independent sequence. And the proof of the Theorem 3.3, 3.4 and 3.5 , we can reference the Zhang and Liang (2013)[19].

Theorem 3.3 Suppose that (A0)(ii), (A1), (A2) and (A4) and (A3)(i)(ii)(iv) for some 1/2 < 𝑠 < 1 are satisfied.

Then 𝑇𝑛2(𝛽̂𝑊₂− 𝛽)/𝛴1𝑛→^𝐷 𝑁(0,1).

Theorem 3.4 Suppose that (A0)(ii), (A1), (A2), (A4) and (A3)(i)(ii)(iv) for some 5/8 < 𝑠 < 1 are satisfied.

Assume that 𝑚𝑎𝑥1|𝑣𝑖| = 𝑂(𝑛^1/3). For each 𝑡 ∈ [0,1], if 𝑛 ∑^𝑛_𝑖=1𝑊_𝑛𝑖²(𝑡) → ∞, then we have [𝑔̂𝑊₂(𝑡) − 𝐸𝑔̂𝑊₂(𝑡)]/𝛤𝑛(𝑡) →^𝐷 𝑁(0,1).

Theorem 3.5 Suppose that (A0)(ii), (A1), (A2), (A4) and (A3)(i)(ii)(iv) for some 𝑠 = 1/2 are satisfied. Assume that 𝑠𝑢𝑝𝑖𝐸𝜇_𝑖⁸< ∞ and 𝑖𝑛𝑓𝑖𝑉𝑎𝑟[(𝜀𝑖− 𝜇𝑖𝛽)²] > 0. For each 𝑢 ∈ [0,1], if 𝑛 ∑^𝑛_𝑖=1𝑊̂_𝑛𝑖²(𝑢) → ∞, then we have [𝑓̂_𝑛(𝑢) − 𝐸𝑓̂_𝑛(𝑢)]/𝛥_𝑛(𝑢) →^𝐷 𝑁(0,1).

4. SIMULATION STUDY

In this section, we carry out a simulation to study the finite sample performance of the proposed estimators. In particular, We examine how good the asymptotic normality is for the estimators of 𝛽, 𝑔(⋅) by Q-Q plot.

Observations are generated from {𝑦_𝑖= 𝜉_𝑖𝛽 + 𝑔(𝑡_𝑖) + 𝜀_𝑖,

𝑥𝑖= 𝜉𝑖+ 𝜇𝑖, 𝑖 = 1,2, ⋯ , 𝑛,

where 𝛽 = 1, 𝑔(𝑡) = sin(2𝜋𝑡), 𝜎_𝑖²= 𝑓(𝑢_𝑖), 𝑓(𝑢) = [1 + 0.5cos(2𝜋𝑢)]², 𝑡_𝑖= (𝑖 − 0.5)/𝑛 and 𝑢_𝑖= (𝑖 − 1)/

𝑛 , 𝜉𝑖= 𝑡𝑖2+ 𝑣𝑖 with 𝑣𝑖= sin(𝑖)/(𝑛^1/3) . {𝜇𝑖, 1} is an i.i.d. 𝑁(0,0. 2²) sequence. { 𝑒𝑖, 1 } are subjected to multivariate normal distribution with 𝐸(𝑒₁, ⋯ , 𝑒_𝑛) = (0, ⋯ ,0) , 𝐶𝑜𝑣(𝑒_𝑖, 𝑒_𝑗) = −4^{−(𝑗−𝑖)−1}𝑓𝑜𝑟𝑖 ≠ 𝑗 and 𝑉𝑎𝑟(𝑒𝑖) = 0. 5²𝑓𝑜𝑟1. For the proposed estimators, the weight functions are taken as

𝑊_𝑛𝑖(𝑡) = ^{𝐾((𝑡−𝑡𝑖)/ℎ𝑛)}

∑^𝑛_𝑗=1𝐾((𝑡−𝑡_𝑗)/ℎ_𝑛), 𝑊̂_𝑛𝑖(𝑢) = ^{𝐾((𝑢−𝑢𝑖)/𝑏𝑛)}

∑^𝑛_𝑗=1𝐾((𝑢−𝑢_𝑗)/𝑏_𝑛). where 𝐾(⋅) is a Gaussian kernel function, ℎ𝑛 and 𝑏𝑛 are two bandwidth sequences.

It is well known that an important issue is the selection of an appropriate bandwidth sequences. This issue has been extensively stuied in the context of nonparametric regression. One of bandwidth selection rules is the delete- one cross-validation rule. It is noted that our estimators may involve two bandwidths. Hence, it is somewhat complicated to select appropriate bandwidths for our estimatos. we state the procedure in the following three steps:

• Select ℎ_𝑛 by minimizing 𝐶𝑉₁(ℎ_𝑛) =¹

𝑛∑^𝑛_𝑖=1(𝑦_𝑖− 𝑥_𝑖𝛽̂_𝐿,−𝑖− 𝑔̂_𝐿,−𝑖(𝑡_𝑖))² where 𝛽̂𝐿,−𝑖 and 𝑔̂𝐿,−𝑖(𝑡𝑖) are "Leave one out" versions of 𝛽̂𝐿 and 𝑔̂𝐿(𝑡𝑖).

• Select ℎ′_𝑛 by minimizing 𝐶𝑉₂(ℎ′_𝑛) =¹

𝑛∑^𝑛_𝑖=1(𝑦_𝑖− 𝑥_𝑖𝛽̂_{𝑊1,−𝑖}− 𝑔̂_{𝑊1,−𝑖}(𝑡_𝑖))² where 𝛽̂𝑊1,−𝑖 and 𝑔̂𝑊1,−𝑖(𝑡𝑖) are "Leave one out" versions of 𝛽̂𝑊1 and 𝑔̂𝑊1(𝑡𝑖).

• Select 𝑏𝑛 by minimizing 𝐶𝑉₃(𝑏_𝑛) =¹

𝑛∑^𝑛_𝑖=1(𝑦_𝑖− 𝑥_𝑖𝛽̂_{𝑊2,−𝑖}− 𝑔̂_{𝑊2,−𝑖}(𝑡_𝑖))² where 𝛽̂_{𝑊2,−𝑖} and 𝑔̂_{𝑊2,−𝑖}(𝑡_𝑖) are "Leave one out" versions of 𝛽̂_𝑊2 and 𝑔̂_𝑊2(𝑡_𝑖).

We found by calculation the corresponding optimal bandwidths ℎ₁= 0.38 and ℎ₂= 0.14.

We give the Q-Q plot for the estimator of 𝛽 and 𝑔(⋅) under the condition that 𝑓(⋅) is known. In Figure 1, we give the Q-Q plot for 𝛽̂𝐿 and 𝛽̂𝑊₁ with 𝑛 = 100,300 and 500, respectively. In Figure 2, we provide the Q-Q plot for 𝑔̂_𝐿(⋅) and 𝑔̂_𝑊1(⋅) with 𝑛 = 100,300 and 500, respectively.

From Figure 1-2, we can see that:

• The asymptotic normality of 𝛽̂_𝐿 or 𝛽̂_𝑊1 is obvious, so does the asymptotic normality of 𝑔̂(⋅);

• The normality becomes more obvious with increasing sample size 𝑛.

Figure 1: The Q-Q plots for 𝛽̂_𝐿 and 𝛽̂_𝑊1 with N=500, n=100,300 and 500, respectively.

Figure 2: The Q-Q plots for 𝑔𝐿(⋅) and 𝑔𝑊₁(⋅) with N=500, n=100,300 and 500, respectively.

5. PRELIMINARY LEMMAS

In the sequel, let 𝑐, 𝑐1, ⋯ and 𝐶, 𝐶1, ⋯ are some finite positive constants, whose values are unimportant and may change. 𝑎_𝑛= 𝑂(𝑏_𝑛) means |𝑎_𝑛||𝑏_𝑛|, while 𝑎_𝑛= 𝑜(𝑏_𝑛) means 𝑎_𝑛/𝑏_𝑛→ 0. 𝑎⁺= max(0, 𝑎), 𝑎⁻= max(0, −𝑎).

And let {𝑒𝑖, 1} be a sequence of zero mean stationary NA random errors. Now, we introduce several lemmas, which will be used in the proof of the main results.

Lemma 5.1 (Baek and Liang (2006)and Baek(2006), Lemma 3.1) Let 𝛼 > 2. Assume that {𝑎𝑛𝑖, 1, 𝑛1} is a triangular array of real numbers with 𝑚𝑎𝑥₁|𝑎_𝑛𝑖| = 𝑂(𝑛^−1/2) and ∑^𝑛_𝑖=1𝑎_𝑛𝑖² = 𝑜(𝑛^−2/𝛼(𝑙𝑜𝑔𝑛)⁻¹). If 𝑠𝑢𝑝_𝑖𝐸|𝑒_𝑖|^𝑝<

∞ for some 𝑝 > 2𝛼/(𝛼 − 1). Then

∑^𝑛_𝑖=1𝑎𝑛𝑖𝑒𝑖= 𝑜(𝑛^−1/𝛼)𝑎. 𝑠.

Remark 5.1 In Lemma 5.1, it is quite clear that 𝑝 > 2 as 𝛼 → ∞ and ∑^𝑛_𝑛𝑖𝑎𝑛𝑖𝑒𝑖= 𝑜(1) a.s.; and 𝑝 > 4 when 𝛼 > 4 and ∑^𝑛_𝑛𝑖𝑎_𝑛𝑖𝑒_𝑖= 𝑜(𝑛^−1/4) a.s. In addition, if all of the "o" is changed into "O", then the conclusion is also right.

Lemma 5.2 (Hardle et al. (2000)(2000), Lemma A.3) Let 𝑉1, ⋯ , 𝑉𝑛 be independent random variables with 𝐸𝑉_𝑖= 0, finite variances and 𝑠𝑢𝑝₁𝐸|𝑉_𝑗|^𝑟< ∞(𝑟 > 2). Assume that {𝑎_𝑘𝑖, 𝑘, 𝑖 = 1, ⋯ , 𝑛} is a sequence of real numbers such that 𝑠𝑢𝑝1,𝑘|𝑎𝑘𝑖| = 𝑂(𝑛^−𝑝1) for some 0 < 𝑝1< 1 and ∑^𝑛_𝑗=1𝑎𝑗𝑖= 𝑂(𝑛^𝑝2) for 𝑝2𝑚𝑎𝑥(0,2/𝑟 − 𝑝1).

Then

max1 | ∑^𝑛_𝑘=1𝑎_𝑘𝑖𝑉_𝑘| = 𝑂(𝑛^−𝑠log𝑛)𝑎. 𝑠. 𝑓𝑜𝑟 𝑠 = (𝑝₁− 𝑝₂)/2.

Lemma 5.3 (Liu and Gan(2003)and Gan(2006)) Assume 𝑎𝑛 is a array of positive real numbers, and

∑^∞_𝑛=1𝜎_𝑛²/𝑎_𝑛²< ∞, where 𝜎_𝑛²= 𝑉𝑎𝑟(𝑒_𝑛). If 0 < 𝑎_𝑛↑ ∞. Then

∑^𝑛_𝑖=1 ^𝑒𝑖

𝑎_𝑛= 𝑜(1)𝑎. 𝑠.

Lemma 5.4 (Han-Ying Liang, Volker Mammitzsch and Josef Steinebach (2006)and Volker(2006), Lemma 4.4(ii)) Let {𝑎𝑛𝑖, 1, 𝑛1} be an array of real numbers and set 𝛥𝑛2 = 𝑉𝑎𝑟(∑^𝑛_𝑖=1𝑎𝑛𝑖𝑒𝑖). Assume that

∑_{𝑗:|𝑘−𝑗|}|𝑐𝑜𝑣(𝑒_𝑘, 𝑒_𝑗)| → 0𝑎𝑠𝑢 → ∞ uniformly for 𝑘1, and 𝑚𝑎𝑥₁|𝑎_𝑛𝑖| = 𝑜(𝛥_𝑛), ∑^𝑛_𝑖=1𝑎_𝑛𝑖² = 𝑂(𝛥_𝑛²). If

∑^𝑛_𝑖=1|𝑎𝑛𝑖| = 𝑂(1), Then

∑^𝑛_𝑖=1^{𝑎𝑛𝑖𝑒𝑖}

Δ_𝑛 →^𝐷 𝑁(0,1).

The proof of the Lemmas 5.5 and 5.6, we can reference the Zhang and Liang (2011)[24] and Zhang and Liang (2013)[19].

Lemma 5.5

• Assumptions (A1), (A2) and (A3), one can imply that 𝑛⁻¹∑^𝑛_𝑖=1𝜉̃_𝑖²→ Σ0, max1|𝜉̃𝑖| = 𝑜(𝑛^−1/2) and 𝑆𝑛−2∑^𝑛_𝑖=1|𝜉̃𝑖|;

• Using (A1), (A2) and (A3), imply that 𝐶₁⁻¹∑^𝑛_𝑖=1𝜎_𝑖⁻²𝜉̃_𝑖²₂ and 𝑇_𝑛⁻²∑^𝑛_𝑖=1|𝜎_𝑖⁻²𝜉̃_𝑖|;

• Let 𝐴̃_𝑖= 𝐴(𝑡_𝑖) − ∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝐴(𝑡_𝑗), where 𝐴(⋅) = 𝑓(⋅), 𝑔(⋅) or ℎ(⋅). Then (A2)(ii) and (A3)(ii) imply that max1|𝐴̃𝑖| = 𝑜(𝑛^−1/4).

Lemma 5.6 Under the condition of Lemma 5.5 and (A0), (A3), we have 𝑆_1𝑛² → 𝑆_𝑛²𝑎. 𝑠.

6. PROOF OF MAIN RESULTS

In the sequel, we use the Abel Inequality (Härdle et al. (2000)[21], page 183). Let 𝐴₁, 𝐴₂, ⋯ , 𝐴_𝑛; 𝐵₁, 𝐵₂, ⋯ , 𝐵_𝑛(𝐵₁₂⋯_𝑛0) to be two sequence of real numbers, and 𝑆_𝑘 = ∑^𝑘_𝑖=1𝐴_𝑖, 𝑀₁= min₁𝑆_𝑘, 𝑀₂= max₁𝑆_𝑘. Then, 𝐵₁𝑀₁∑^𝑛_𝑘=1𝐴_𝑘𝐵_𝑘1𝑀₂. Let 𝐸_𝑖, 𝐹_𝑖(1) to be arbitrary real numbers and (𝑗₁, 𝑗₂, ⋯ , 𝑗_𝑛) to be a permutation of (1, ⋯ , 𝑛) such that 𝐹𝑗₁𝑗₂⋯𝑗_𝑛. Then from the above equation, we have

| ∑^𝑛_𝑖=1𝐸𝑖𝐹𝑖| = | ∑^𝑛_𝑖=1𝐸𝑗_𝑖𝐹𝑗_𝑖|| ∑^𝑛_𝑖=1𝐸𝑗_𝑖(𝐹𝑗_𝑖− 𝐹𝑗_𝑛)| + | ∑^𝑛_𝑖=1𝐸𝑗_𝑖𝐹𝑗_𝑛| 𝐶max1 |𝐹_𝑖|max

1 | ∑^𝑚_𝑖=1𝐸_𝑗𝑖|. (1)

Proof of Theorem 3.1. We prove only (a), as the proof of (b) is analogous. From (1) and (??), write that 𝛽̂𝐿− 𝛽 = 𝑆1𝑛−2[∑^𝑛_𝑖=1(𝜉̃𝑖+ 𝜇̃𝑖)(𝑦̃𝑖− 𝜉̃𝑖𝛽 − 𝜇̃𝑖𝛽) + 𝑛Ξ𝜇2𝛽]

= 𝑆_1𝑛⁻²{∑^𝑛_𝑖=1[(𝜉̃_𝑖+ 𝜇̃_𝑖)(𝜀̃_𝑖− 𝜇̃_𝑖𝛽) + Ξ_𝜇²𝛽] + ∑^𝑛_𝑖=1𝜉̃_𝑖𝑔̃_𝑖+ ∑^𝑛_𝑖=1𝜇̃_𝑖𝑔̃_𝑖}

= 𝑆1𝑛−2{∑^𝑛_𝑖=1[(𝜉̃𝑖+ 𝜇𝑖)(𝜀𝑖− 𝜇𝑖𝛽) + Ξ𝜇2𝛽] + ∑^𝑛_𝑖=1𝜉̃𝑖𝑔̃𝑖+ ∑^𝑛_𝑖=1𝜇̃𝑖𝑔̃𝑖

+ ∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝜉̃_𝑖𝜇_𝑗𝛽 − ∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝜉̃_𝑖𝜀_𝑗− ∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝜀_𝑖𝜇_𝑗

− ∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝑊𝑛𝑗(𝑡𝑖)𝜇𝑖𝜀𝑗+ 2 ∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝑊𝑛𝑗(𝑡𝑖)𝜇𝑖𝜇𝑗𝛽

+ ∑^𝑛_𝑖=1∑^𝑛_𝑗=1∑^𝑛_𝑘=1𝑊𝑛𝑗(𝑡𝑖)𝑊𝑛𝑘(𝑡𝑖)𝜇𝑗𝜀𝑘− ∑^𝑛_𝑖=1∑^𝑛_𝑗=1∑^𝑛_𝑘=1𝑊𝑛𝑗(𝑡𝑖)𝑊𝑛𝑘(𝑡𝑖)𝜇𝑗𝜇𝑘𝛽}

: = 𝑆_1𝑛⁻²∑¹⁰_𝑙=1𝐴_𝑙𝑛. (2)

Thus. Using Lemma 5.6, in order to prove 𝑆_𝑛²(𝛽̂_𝐿− 𝛽)/Σ_𝑛→^𝐷 𝑁(0,1), we verify that

𝐴_1𝑛

Σ_𝑛 →^𝐷 𝑁(0,1)^𝐴𝑘𝑛

Σ_𝑛 →^𝑃 0𝑓𝑜𝑟𝑘 = 2,3,4,5,8,10.^𝐴𝑘𝑛

Σ_𝑛 →^𝑃 0𝑓𝑜𝑟𝑘 = 6,7,9.

Step 1. we prove that 𝐴_1𝑛/Σ_𝑛→^𝐷 𝑁(0,1).

Set 𝜔_𝑖= (𝜉̃_𝑖+ 𝜇_𝑖)(𝜀_𝑖− 𝜇_𝑖𝛽) + Ξ_𝜇²𝛽 and 𝑍_𝑛𝑖= 𝜔_𝑖/Σ_𝑛. According to Zhang and Liang (2013)[19] we have Σ_𝑛²𝐶𝑛. Using (A0), Lemma 5.5, Σ_𝑛²𝐶𝑛. We deduce that 𝐸𝑍_𝑛𝑖= 0, V𝑎𝑟(∑^𝑛_𝑖=1𝑍_𝑛𝑖) = 1 and 𝐸|𝑍_𝑛𝑖|^2+𝛿< ∞. Owing to {𝜀_𝑖} are sequence of zero mean stationary NA random variables, {𝑒_𝑖𝜎_𝑖− 𝜇_𝑖𝛽} are also sequence of zero mean stationary NA random variables, {𝜉̃_𝑖+ 𝜇_𝑖} are sequence of i.i.d. random variables. Using Definition (1), we know (𝜉̃𝑖+ 𝜇𝑖)(𝜀𝑖− 𝜇𝑖𝛽) are sequence of 𝜌-mixing random variables, and the mixing coefficients 𝜌(𝑛) = 0. In this situation, we can know 𝜌-mixing is also a squence of strong mixing from Fan and Yao (2003)[25], and we have 0𝛼(𝑛)𝜌(𝑛)/4 = 0. Therefore, (𝜉̃_𝑖+ 𝜇_𝑖)(𝜀_𝑖− 𝜇_𝑖𝛽) are sequences of strong mixing random variables with the mixing coefficients 𝛼(𝑛) = 0. Thus. By the proof of Theorem 3.1 of Zhang and Liang (2013)[19], we accept the conclusion as correct.

Step 2. We prove that 𝐴𝑘𝑛/Σ𝑛→ 0 for 𝑘 = 2,3,4,5,8,10.

From (A0)(i), (A3) and Lemma 5.2, we can verity that

∑^𝑛_𝑖=1(𝜁_𝑖− 𝐸𝜁_𝑖) = 𝑂(𝑛¹²𝑙𝑜𝑔𝑛)𝑎. 𝑠. max

1 | ∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝜇_𝑗| = 𝑂(𝑛⁻¹⁴𝑙𝑜𝑔𝑛)𝑎. 𝑠. (3) where 𝜁_𝑖= |𝜇_𝑖|, 𝜇_𝑖²𝑜𝑟𝜇_𝑖.

Since the {𝜀_𝑖, 𝑖 = 1,2, . . . , 𝑛} are sequence of zero mean stationary NA random errors, the {𝜀_𝑖⁺, 𝑖 = 1,2, . . . 𝑛} and {𝜀_𝑖⁻, 𝑖 = 1,2, . . . 𝑛} are all NA sequence. From Lemma 5.3, one can get 1/𝑛 ∑^𝑛_𝑖=1𝜀_𝑖⁺= 𝑜(1)𝑎. 𝑠. ,1/

𝑛 ∑^𝑛_𝑖=1𝜀_𝑖⁻= 𝑜(1)𝑎. 𝑠. And |𝜀_𝑖| = 𝜀_𝑖⁺+ 𝜀_𝑖⁻, we have

1

𝑛∑^𝑛_𝑖=1|𝜀_𝑖| = 𝑜(1)𝑎. 𝑠. (4)

Hence, by applying (A0)(i) and (A3), Lemma 5.1 and 𝑎_𝑛= 𝑊_𝑛𝑗(𝑡_𝑖), 𝛼 = 4, one can obtain that

max1 | ∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝜀_𝑗| = 𝑜(𝑛⁻¹⁴)𝑎. 𝑠. (5)

So. From (A0)(i), (A1), (A2), (A3), Lemma 5.5, (??), (3), (5) we deduce that

|^𝐴2𝑛

Σ_𝑛 | ^𝐶

√𝑛| ∑^𝑛_𝑖=1𝜉̃_𝑖𝑔̃_𝑖| ^𝐶

√𝑛{| ∑^𝑛_𝑖=1ℎ̃_𝑖𝑔̃_𝑖| + | ∑^𝑛_𝑖=1𝑣_𝑖𝑔̃_𝑖| + | ∑^𝑛_𝑖=1[∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝑣_𝑗]𝑔̃_𝑖|}

𝐶

√𝑛[𝑛 ⋅ max

1 |ℎ̃𝑖| ⋅ max

1 |𝑔̃𝑖| + max

1 |𝑔̃𝑖| ⋅ max

1 | ∑^𝑛_𝑖=1𝑣𝑘_𝑖| +max1 ∑^𝑛_𝑖=1𝑊_𝑛𝑗(𝑡_𝑖) ⋅ max

1 |𝑔̃_𝑖| ⋅ max

1 | ∑^𝑛_𝑗=1𝑣_𝑘𝑗|]

= 𝑜(1) + 𝑜(𝑛^−1/4log𝑛) = 𝑜(1).

𝐸(^𝐴3𝑛

Σ_𝑛)^{2 𝐶}

𝑛{𝐸(∑^𝑛_𝑖=1𝑔̃_𝑖𝜇_𝑖)²+ 𝐸[∑^𝑛_𝑖=1𝑔̃_𝑖∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝜇_𝑗]²}

=^𝐶

𝑛{∑^𝑛_𝑖=1𝑔̃_𝑖²+ ∑^𝑛_𝑗=1[∑^𝑛_𝑖=1𝑊_𝑛𝑗(𝑡_𝑖)𝑔̃_𝑖]²} = 𝑜(𝑛^−1/2)

𝐴_4𝑛= ∑^𝑛_𝑖=1ℎ̃_𝑖∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝜇_𝑗𝛽 + ∑^𝑛_𝑖=1𝑣_𝑖∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝜇_𝑗𝛽 −

∑^𝑛_𝑖=1∑^𝑛_𝑠=1𝑊_𝑛𝑠(𝑡_𝑖)𝑣_𝑠∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝜇_𝑗𝛽 : = 𝐷1𝑛+ 𝐷2𝑛+ 𝐷3𝑛. 𝐸(^𝐷1𝑛

Σ_𝑛)^{2 𝐶}

𝑛∑^𝑛_𝑗=1[∑^𝑛_𝑖=1𝑊𝑛𝑗(𝑡𝑖)ℎ̃𝑖]²⋅ max

1 |ℎ̃𝑖|²⋅ max

1 | ∑^𝑛_𝑖=1𝑊𝑛𝑗(𝑡𝑖)|²= 𝑜(𝑛^−1/2).

𝐸(^𝐷2𝑛

Σ_𝑛)^{2 𝐶}

𝑛∑^𝑛_𝑗=1[∑^𝑛_𝑖=1𝑊_𝑛𝑗(𝑡_𝑖)𝑣_𝑖]²⋅ max

1 | ∑^𝑚_𝑖=1𝑣_𝑘𝑖|²⋅ max

1,𝑗 𝑊_𝑛𝑗²(𝑡_𝑖) = 𝑜(1).

|𝐷_3𝑛| Σ_𝑛

𝐶

√𝑛max

1 | ∑^𝑚_𝑠=1𝑣_𝑘𝑠| ⋅ max

1 ∑^𝑛_𝑖=1𝑊_𝑛𝑠(𝑡_𝑖) ⋅ max

1 | ∑^𝑛_𝑗=1𝑊_𝑛𝑗(𝑡_𝑖)𝜇_𝑗| = 𝑜_𝑝(1).