# Averaged and Integrated Estimations of Varying-Coefficient Regression Models with Dependent Observations

Authors: Guo-Liang Fan; Hong-Xia Xu
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

In practical applications, lots of data such as sequentially collected economic data often exhibit some evident dependence. This paper studies the varying-coefficient regression models with different smoothing variables when the data form a stationaryα-mixing sequence. Both the averaged and integrated estimators of coefficient functions are proposed. The asymptotic normalities of the proposed averaged and integrated estimators are also established.


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## Body

## 1. Introduction

Regression analysis is one of the most mature and widely applied branches of statistics. Various regression models have been studied by many authors; e.g., Liang and Fan [1] studied Berry-Esseen type bounds of estimators in a semiparametric model with linear process errors; Fan, Liang, and Xu [2] considered empirical likelihood confidence regions for heteroscedastic partial linear model and so on. Recently, varying models which were proposed by Hastie and Tibshirani [3] and Chen and Tsay [4] have received more and more attention. This is mainly because varying model offer a flexible but parsimonious alternative to nonparametric model and has been used in many contexts such as it has been successfully applied to multidimensional nonparametric regression, generalized linear model, time series analysis, longitudinal and functional data analysis, and time-varying model in finance. A nice feature of varying coefficients is to allow appreciable flexibility on the structure of fitted models without suffering from the “curse of dimensionality”.Consider the following varying-coefficient model with different smoothing variables in different coefficient functions:(1)Y=∑α=1paαXαZα+σX,Zε,where (Y,Xτ,Zτ) is a random vector, X=(X1,…,Xp)τ∈Rp and Z=(Z1,…,Zp)τ∈Rp; {aα(·)}α=1p are some measurable functions from R to R; σ·,· is a measurable function from R2p to R and ε is independent of (X,Z) with mean zero and variance 1.Most of the work focused on the case where all coefficient functions share a single smoothing variable in a model, i.e.X1=X2=⋯=Xp in model (1). In particular, the asymptotic normalities of the estimators of these coefficient functions by local polynomial technique are obtained, such as Shen et al. [5], Wang and Lin [6], and Fan, Liang, and Zhu [7]. However, the varying-coefficient regression models on which different coefficient functions share different variables have been given less attention because the local polynomial technique is not adequate. To solve this difficulty, when (Xi,Zi,εi), which come from (X,Z,ε), are independent and identically distributed (i.i.d.), Zhang and Li [8] employed the local linear technique to obtain an initial value for the estimate of each coefficient function in model (1), and, then, the integrated estimate of each coefficient function is defined by integrating its initial value on these variables which the coefficient function does not share; Zhang and Li [9] used the local linear technique to give an initial value for every estimate of the coefficient function in model (1). Then the averaged estimate of each coefficient function is defined by averaging its initial value on these variables which the coefficient function does not share; Yang [10] proposed estimators for this model under random right censoring case by using mean-preserving transformation and established their asymptotic properties. The estimation procedure is based on the profiling and the smooth backfitting techniques.However, the above related papers need the i.i.d. assumption for the data. The independence assumptions are not always valid in applications, especially for sequentially collected economic data, which often exhibit evident dependence. In the sequel,(Xi,Zi,εi) are assumed to form a strong stationary α-mixing sequence. Fan et al. [11] studied the varying-coefficient errors-in-variables models when the data form a stationary α-mixing sequence of random variables. Recall that a sequence {ζi,i≥1} is said to be α-mixing if the α-mixing coefficient (2)αm≔supk≥1⁡sup⁡PAB-PAPB:A∈Fm+k∞,B∈F1kconverges to zero as m→∞, where Flm=σζl,ζl+1,…,ζm denotes the σ-algebra generated by ζl,ζl+1,…,ζm with l≤m. Among various mixing conditions used in the literature, α-mixing is reasonably weak and has many practical applications. In fact, under very mild assumptions linear autoregressive and more generally bilinear time series models are α-mixing with mixing coefficients decaying exponentially; i.e., α(k)=O(ρk) for some 0<ρ<1; see Doukhan [12], page 99, for more details.The paper is organized as follows. In Section2, we list some assumptions and present the averaged and integrated estimators of the coefficient functions, as well as the asymptotical normalities for these estimators. Proofs of the main results and the preliminary lemmas are provided in Section 3. Concluding remarks are given in Section 4.


## 2. Methodology and Main Results

Suppose that{Yi,Xi,Zi,i=1,…,n} is a sample from model (1), (Xi,Zi,εi) are strong stationary α-mixing sequence, and that for every α=1,…,p, aα(·) has a Lipschitz continuous second derivative. Then aα(Xα) can be approximate locally by a linear function, aα(Xα)≈aα(xα)+aα′(xα)(Xα-xα), at a neighborhood of xα which is in the support of Xα. We minimize(3)∑i=1nYi-∑α=1paαZiα-∑α=1pbαXiα-xαZiα2∏α=1pKα,hαXiα-xαwith respect to {aα}α=1p and {bα}α=1p, where Xiα and Ziα are the αth entries of the ith observation Xi and Zi, respectively; Kα,hα(·)=hα-1Kα(·/hα), Kα(·) is a bounded, nonnegative, compactly supported symmetric about zero and Lipschitz continuous density function; hα=hnα is a sequence of positive numbers and called a bandwidth.Leta~α(x),α=1,…,p, be the first p entries of the minimizers of (3); then, it follows from the least squares theory that(4)a~αx=∑i=1neα,2pτUτWU-1UiWiiYi,α=1,…,p,where x=(x1,…,xp)τ,eα,2p is a unite vector with 1 at its αth position and U is an n×2p matrix with Uiτ=(Zi1,…,Zip,((Xi1-x1)/h1)Zi1,…,((Xip-xp)/hp)Zip) as its ith row, W=diag⁡(W11,…,Wnn) with Wii=∏α=1pKα,hα(Xiα-xα). a~α(x) is called as a initial value of the estimate of the coefficient function aα(x).The averaged estimate of the coefficient functionaα(xα) is defined as (see [9])(5)a^αxα=1n∑j=1na~αXj1,…,Xj,α-1,xα,Xj,α+1,…,Xjp,α=1,…,p.Next, we define the integrated estimate of the coefficient function. LetQ-α(x-α) be a deterministic weight function with ∫dQ-α(x-α)=1, for each α=1,…,p, where x-α=(x1,…,xα-1,xα+1,…,xp)τ. We allow for both discrete and continuous Q-α(x-α) and integrals should be interpreted in the Stieltjes sense. Let q-α(x-α) be the density of Q-α(x-α) with respect to either Lebesgue or a counting measure. Suppose that the support of Q-α(·) is contained within that of x-α. Then, the integrated estimate of the coefficient function aα(xα) is defined as (see [8])(6)a˘αxα=∫a~αxdQ-αx-α,α=1,…,p.In order to formulate the main results, we need the following assumptions.(A1)
(i)
The joint densityp(x) of X and the marginal density pα(xα) of Xα, α=1,…,p, are compactly supported, bounded, Lipschitz continuous, and bounded away from zero by a constant.(ii)
The conditional probabilities satisfyP(Xi,Xj∣Xk)=P(Xi∣Xk)·P(Xj∣Xk) for different 1≤i,j,k≤n.(A2)
The conditional expectsE(ZαZα′σ2(X,Z)∣X=x) and E(Zαλ1Zα′λ2∣X=x) are Lipschitz continuous, λ1+λ2=2 or 4, λ1≥0,λ2≥0,α,α′=1,…,p.(A3)
E(ε+Z+X)2+δ<∞, for some δ>0.(A4)
(i)
α(n)=O(n-γ), for some γ>1+2/δ.(ii)
There exist positive integersr≔r(n) and q≔q(n) such that for sufficiently large n, r+q≤n, nα(q)/r=o(1) and rλ/2=on5λ+2λδ-δ/52+δ, for some 0<λ<δ.Remark 1.
Conditions (A1)(i) and (A2) have been assumed in Zhang and Li [8, 9]. (A1)(ii) holds naturally when Xi are i.i.d. (A3) has been used by Fan et al. [2]. Assumption (A4)(i) is a mild condition. In the particular case of exponential decay, α(k)=O(ρk) for some 0<ρ<1, we have that α(k)=O(k-γ) for sufficient large γ, and hence assumption (A3) is satisfied. The technical condition (A4) is very easy to be satisfied. For example, when δ=1 and α(n)=O(n-4), choosing r=[n1/2], q=[n1/6], and λ=1/2, we have nα(q)/r=O(n-1/6) and rλ/2/n5λ+2λδ-δ/5(2+δ)=O(n-1/24).Theorem 2.
Suppose that (A1)–(A4) hold and the bandwidths are such thatn∏α=1phα→∞, and for each α=1,…,p, hα=Cαn-1/5, and nhα′5→0, α′≠α, α′=1,…,p, Cα is a constant; then(7)nhαa^αxα-aαxα-12μ2Kαaα′′xhα2→dN0,v0Kαvα2xα,where μλ(Kα)=∫uλKα(u)du, vλ(Kα)=∫uλKα2(u)du, vα2(xα)=∫(p-α2(x-α)/p(x))E[{∑α′=1pSαα′(x)Zα′}2σ2(X,Z)∣X=x]dx-α, S(x)=E(ZZτ∣X=x)={Sλ1,λ2(x)}1≤λ1,λ2≤p, and its inverse S-1(x)={Sλ1,λ2(x)}1≤λ1,λ2≤p, p-α(x-α) is the marginal density of X-α=(X1,…,Xα-1,Xα+1,…,Xp)τ. In particular, when σ(X,Z) is a constant σ, vα2(xα)=σ2∫{p-α2(x-α)/p(x)}·Sαα(x)dx-α.Theorem 3.
Under the conditions of Theorem2, we have(8)nhαa˘αx-aαxα-12μ2Kαaα′′xhα2→dN0,v0Kαv˘α2xα,where v˘α2(xα)=∫q-α2(x-α)/p(x)E[{∑α′=1pSαα′(x)Zα′}2σ2(X,Z)∣X=x]dx-α. In particular, when σ(X,Z) is a constant σ, v˘α2(xα)=σ2∫{q-α2(x-α)/p(x)}Sαα(x)dx-α.


## 3. Proof of the Main Results

In this section, we list some preliminary Lemmas. Let{Zi,i≥1} be a stationary α-mixing sequence of random variables with mixing coefficients {α(m)}. In the sequel, let C denote positive constant whose value may vary at each occurrence. In case of no loss of generalization and expositional purpose, we shall work with the special case p=2.Lemma 4 (see [13]).
LetV1,…,Vn be α-mixing random variables measurable with respect to the σ-algebra Fi1j1,…,Fimjm, respectively, with 11≤i1<j1<⋯<jm≤n,il+1-jl≥w≥1 and Vj≤1 for l,j=1,2,…,m. Then (9)E∏j=1mVj-∏j=1mEVj≤16m-1αw,where Fab=σVi,a<i≤b denotes σ-field generated by Va+1,Va+2,…,Vb, αn is the mixing coefficient.Lemma 5 (see [14], Corollary  A.2, p. 278).
Suppose thatX and Y are random variables such that EXs<∞,EYt<∞, where s,t>1,s-1+t-1<1. Then (10)EXY-EXEY≤8XsYtsupA∈σX,B∈σY⁡PAB-PAPB1-s-1-t-1.Lemma 6 (see [15], Theorem  4.1).
Let2<s<t≤∞,2<k≤t and EZn=0. Assume that α(n)≤Cn-γ for some C>0 and γ>0. If γ>k/(k-2) and γ≥(s-1)t/(t-s). then, for any ϵ>0, there exists Q=Q(ϵ,s,t,k,γ,C)<∞ such that (11)E∑i=1nZis≤Qns/2max1≤i≤n⁡Ziks+n1+ϵmax1≤i≤n⁡Zits.Lemma 7 (see [16], Lemma  2.3).
Assumeα(k)≤C1k-r, for some r>1,C1>0. Let sup1≤i,j≤n,i≠j⁡covZi,Zj≔R∗(n)<∞ be satisfied. Moreover, let Rm(n)<∞ for some m, 2r/(r-1)<m≤∞, where Rm(n)=sup1≤i≤n⁡(EZim)1/m, for 1≤m<∞, and R∞(n)=sup1≤i≤n⁡esssupw∈Ω⁡Zi. Then (12)var∑i=1nZi≤nC2r,mRmn2m/rm-2R∗n1-m/rm-2+R22nholds with C2(r,m)≔20r-40r/m/r-1-2r/mC11/r.Lemma 8.
If assumptions (A1)–(A4) hold andh1=Cn-1/5, nh25→0, then (13)Snx=1nUτWU=pxS0x1+op1andSn-1x=p-1xS0-1x1+op1,in the sense that each element converges, where(14)S0x=EZ12∣X=xEZ1Z2∣X=x00EZ1Z2∣X=xEZ22∣X=x0000μ2K1EZ12∣X=x0000μ2K2EZ22∣X=x.Proof.
Let(15)Sλ1,λ2,r1,r2x=1n∑l=1nZlλ1Zlλ2Xlλ1-xλ1hλ1r1Xlλ2-xλ2hλ2r2K1,h1Xlλ1-xλ1K2,h2Xlλ2-xλ2;then Sn(x) can be presented (16)Snx=S1,1,0,0xS1,2,0,0xS1,1,1,0xS1,2,0,1xS1,2,0,0xS2,2,0,0xS1,2,1,0xS2,2,0,1xS1,1,1,0xS1,2,1,0xS1,1,2,0xS1,2,1,1xS1,2,0,1xS2,2,0,1xS1,2,1,1xS2,2,0,2x.Note that (Xi,Zi) is a strong stationary sequence of α-mixing random variables; then we have(17)ESλ1,λ2,r1,r2x=EEZλ1Zλ2∣X=xXλ1-xλ1hλ1r1Xλ2-xλ2hλ2r2K1,h1Xλ1-xλ1K2,h2Xλ2-xλ2=pxEZλ1Zλ2∣X=x1+o1r1=r2=00r1=1orr2=1pxEZλ1Zλ2∣X=xμr1Kλ1μr2Kλ21+o1r1=0,r2=2orr1=2,r2=0.
SetMl=Zlλ1Zlλ2Xlλ1-xλ1/hλ1r1Xlλ2-xλ2/hλ2r2K1(Xlλ1-xλ1/hλ1)K2(Xlλ2-xλ2/hλ2). It is easy to see (18)sup1≤l≤n⁡EMlm1/m=Ohλ1hλ2,for1≤m<∞,andsup1≤i,j≤n,i≠j⁡covMi,Mj=Ohλ12hλ22,Which, combing with Lemma 7, (A4) and h1=Cn-1/5, nh25→0, yield that, for m>2,(19)varSλ1,λ2,r1,r2x≤Cnn2h12h22hλ1hλ22m/rm-2hλ12hλ221-m/rm-2+hλ1hλ22=Ohλ12hλ22nh12h22=o1.Conjoining (17) and (19), we get the result of this lemma.Proof of Theorem2.
Note thata~1(x) and a~2(x) can be represented that (20)a~1x=1n∑i=1np-1xS11xZi1+S12xZi2K1,h1Xi1-x1K2,h2Xi2-x2Yi1+op1,a~2x=1n∑i=1np-1xS12xZi1+S22xZi2K1,h1Xi1-x1K2,h2Xi2-x2Yi1+op1.Since (21)a^1x1=1n2∑j=1n∑i=1n1px1,Xj2S11x1,Xj2Zi1+S12x1,Xj2Zi2K1,h1Xi1-x1K2,h2Xi2-Xj2×Yi1+op1,a1x1=1n∑j=1na1x1=1n∑j=1ne1,4τUτWU-1UτWUa1x1,a2Xj2,h1a1′x1,h2a2′Xj2τ=1n2∑j=1n∑i=1n1px1,Xj2S11x1,Xj2Zi1+S12x1,Xj2Zi2K1,h1Xi1-x1K2,h2Xi2-Xj2×wix1,Xj21+op1,where wi(x1,Xj2)=a1(x1)Zi1+a2(Xj2)Zi2+a1′(x1)(Xi1-x1)Zi1+a2′(Xj2)(Xi2-x2)Zi2. Then we can obtain that (22)a^1x1-a1x1=A11+A12+A21+op1,where (23)A11=1n2∑j=1n∑i=1n1px1,Xj2S11x1,Xj2Zi1+S12x1,Xj2Zi2K1,h1Xi1-x1K2,h2Xi2-Xj2×a1Xi1Zi1-a1x1Zi1-a1′x1Xi1-x1Zi1,A12=1n2∑j=1n∑i=1n1px1,Xj2S11x1,Xj2Zi1+S12x1,Xj2Zi2K1,h1Xi1-x1K2,h2Xi2-Xj2×a2Xi2Zi2-a2Xj2Zi2-a2′Xj2Xi2-x2Zi2,A2=1n2∑j=1n∑i=1n1px1,Xj2S11x1,Xj2Zi1+S12x1,Xj2Zi2K1,h1Xi1-x1K2,h2Xi2-Xj2×σXi,Ziεi.
Next, we establish that(24)1n∑j=1n1px1,Xj2S11x1,Xj2Zi1+S12x1,Xj2Zi2K2,h2Xi2-Xj2=p2Xi2px1,Xi2S11x1,Xi2Zi1+S12x1,Xi2Zi21+op1.In order to prove (24), we need only to show (25)E1n∑j=1n1px1,Xj2S11x1,Xj2Zi1+S12x1,Xj2Zi2K2,h2Xi2-Xj2-E1n∑j=1n1px1,Xj2S11x1,Xj2Zi1+S12x1,Xj2Zi2K2,h2Xi2-Xj2∣Xi2,Zi1,Zi22→0.In fact, according to (A1)(ii), (A3), and Lemma 5, we have(26)E1n∑j=1n1px1,Xj2S11x1,Xj2Zi1+S12x1,Xj2Zi2K2,h2Xi2-Xj2-E1n∑j=1n1px1,Xj2S11x1,Xj2Zi1+S12x1,Xj2Zi2K2,h2Xi2-Xj2∣Xi2,Zi1,Zi22=1n2∑j1=1n∑j2=1nE1px1,Xj12S11x1,Xj12Zi1+S12x1,Xj12Zi2K2,h2Xi2-Xj12-E1px1,Xj12S11x1,Xj12Zi1+S12x1,Xj12Zi2K2,h2Xi2-Xj12∣Xi2,Zi1,Zi2·1px1,Xj22S11x1,Xj22Zi1+S12x1,Xj22Zi2K2,h2Xi2-Xj22-E1n∑j=1n1px1,Xj22S11x1,Xj22Zi1+S12x1,Xj22Zi2K2,h2Xi2-Xj22∣Xi2,Zi1,Zi2=1n2∑j1=1n∑j2=1nEE1px1,Xj12S11x1,Xj12Zi1+S12x1,Xj12Zi2K2,h2Xi2-Xj12-E1px1,Xj12S11x1,Xj12Zi1+S12x1,Xj12Zi2K2,h2Xi2-Xj12∣Xi2,Zi1,Zi2·1px1,Xj22S11x1,Xj22Zi1+S12x1,Xj22Zi2K2,h2Xi2-Xj22-E1n∑j=1n1px1,Xj22S11x1,Xj22Zi1+S12x1,Xj22Zi2K2,h2Xi2-Xj22∣Xi2,Zi1,Zi2∣Xi2,Zi1,Zi2=1n2∑j1=1n∑j2=1nEE1px1,Xj12S11x1,Xj12Zi1+S12x1,Xj12Zi2K2,h2Xi2-Xj12·1px1,Xj22S11x1,Xj22Zi1+S12x1,Xj22Zi2K2,h2Xi2-Xj22∣Xi2,Zi1,Zi2-E1px1,Xj12S11x1,Xj12Zi1+S12x1,Xj12Zi2K2,h2Xi2-Xj12∣Xi2,Zi1,Zi2·E1px1,Xj22S11x1,Xj22Zi1+S12x1,Xj22Zi2K2,h2Xi2-Xj22∣Xi2,Zi1,Zi2=1n2∑j1=1n∑j2=1nEE1px1,Xj12S11x1,Xj12Zi1+S12x1,Xj12Zi2K2,h2Xi2-Xj12·1px1,Xj22S11x1,Xj22Zi1+S12x1,Xj22Zi2K2,h2Xi2-Xj22·PXj12Xj22∣Xi2,Zi1,Zi2-E1px1,Xj12S11x1,Xj12Zi1+S12x1,Xj12Zi2K2,h2Xi2-Xj12·E1px1,Xj22S11x1,Xj22Zi1+S12x1,Xj22Zi2K2,h2Xi2-Xj22·PXj12∣Xi2,Zi1,Zi2·PXj22∣Xi2,Zi1,Zi2=1n2∑j1=1n∑j2=1nEPXj12Xj22∣Xi2,Zi1,Zi2·E1px1,Xj12S11x1,Xj12Zi1+S12x1,Xj12Zi2K2,h2Xi2-Xj12·1px1,Xj22S11x1,Xj22Zi1+S12x1,Xj22Zi2K2,h2Xi2-Xj22-E1px1,Xj12S11x1,Xj12Zi1+S12x1,Xj12Zi2K2,h2Xi2-Xj12·E1px1,Xj22S11x1,Xj22Zi1+S12x1,Xj22Zi2K2,h2Xi2-Xj22≤Cn-2∑j1=1n∑j2=1nαδ/2+δj2-j1≤C1n2∑j1=1n∑k=1∞αδ/2+δk≤Cn-1→0.Thus, (24) holds.
Then similar to the proof of Theorem in Zhang and Li [9], we have (27)A11=12μ2K1a1′′x1h121+op1andA12=Oph22.Hence, in order to prove Theorem 2, we need only to show that(28)nh1A2→dN0,v0K1v12x1.
DenoteDi=h1p2(Xi2)/p(x1,Xi2)(S11(x1,Xi2)Zi1+S12(x1,Xi2)Zi2)K1,h1(Xi1-x1)σ(Xi,Zi)εi. Then A2 can be represented as (29)A2=1nh1∑i=1nDi1+op1.In order to prove (28), we need only to show(30)1n∑i=1nDi→dN0,v0K1v12x1.
Letηmn,ηmn′,ηwn′ be defined as follows: (31)ηmn=∑i=kmkm+r-1Di,ηmn′=∑j=lmlm+q-1Dj,ηwn′′=∑k=wr+q+1nDk,where km=(m-1)(r+q)+1, lm=(m-1)(r+q)+r+1,m=1,…,w. Then (32)n-1/2∑i=1nDi=n-1/2∑m=1wηmn+∑m=1wηmn′+ηwn′′≔n-1/2Sn′+Sn′′+Sn′′′.Hence, in order to prove (30), it suffices to show that(33)1nESn′′2→0,1nESn′′′2→0,(34)varn-1/2Sn′→v0K1v12x1,(35)Eexp⁡it∑m=1wηmn-∏m=1wEexp⁡itηmn→0,(36)1n∑m=1wEηmn2Iηmn>ϵn→0,∀ϵ>0.Relation (33) implies that Sn′′ and Sn′′′ are asymptotically negligible, (35) shows that the summands ηmn in Sn′ are asymptotically independent, and (34) and (36) are the standard Lindeberg-Feller conditions for asymptotic normality of Sn′ under independence.
We first establish (33). Obviously (37)ESn′′2=∑m=1w∑i=lmlm+q-1EDi2+2∑m=1w∑lm≤i<j≤lm+q-1covDi,Dj+2∑1≤i<j≤wcovηin′,ηjn′≔J1n+J2n+J3n.Note that(38)EDi2=h1Ep22X2p2x1,X2S11x1,X2Z1+S12x1,X2Z22K1,h12X1-x1σ2X,Z=h1Ep22X2p2x1,X2ES11x1,X2Z1+S12x1,X2Z22σ2X,Z∣X=xK1,h12X1-x1=v0K1∫p22X2px1,x2ES11x1,x2Z1+S12x1,x2Z22σ2X,Z∣X=xdx21+o1=v0K1v12x11+o1,which yields that J1n=O(wq/n)=o(1) from (A4). Since (39)J2n≤2n∑1≤i<j≤ncovDi,DjandJ3n≤2n∑1≤i<j≤ncovDi,Dj,to prove J2n=o(1) and J3n=o(1), it suffices to show that(40)1n∑1≤i<j≤ncovDi,Dj→0.Next, let cn (specified below) be a sequence of integers such that cn→∞ and cnh1→0. Put (41)S1=i,j∣i,j∈1,2,…,n,1≤j-i≤cnS2=i,j∣i,j∈1,2,…,n,cn+1≤j-i≤n-1.We write(42)1n∑1≤i<j≤ncovDi,Dj=1n∑S1covDi,Dj+1n∑S2covDi,Dj.It is easy to see that (43)covDi,Dj≤EDiDj+EDiEDj=Oh1.Hence,(44)1n∑S1covDi,Dj=Ocnh1→0.On the other hand, it follows from Lemma 5 that (45)covDi,Dj≤Cαj-i1-2/2+δEDi2+δ2/2+δand(46)EDi2+δ=h11+δ/2Ep22+δX2p2+δx1,X2S11x1,X2Z1+S12x1,X2Z22+δK1,h12+δX1-x1σ2+δX,Z=Oh1-δ/2.Then we have (47)1n∑S2covDi,Dj≤Cn∑j=1n∑j-i=cn+1n-1αj-i1-2/2+δh1-δ/2+δ≤Ch1-δ/2+δ∑l=cn∞αlδ/2+δ≤Ccn-δ1h1-δ/2+δ∑l=cn∞lδ1αlδ/2+δ.Therefore, by choosing cn=h1-δ/((2+δ)δ1) and in view of (A4), we obtain that(48)1n∑S1covDi,Dj=Ocnh1→0.Thus, (40) is verified from (42), (44), (46), and (48).
As ton-1E(Sn′′′)2, by (A4), (38), (40), and Lemma 6, we have (49)1nESn′′′2≤1n∑i=wr+q+1nEDi2+2n∑wr+q+1≤i<j≤ncovDi,Dj≤Cn-wp+qn+2n∑1≤i<j≤ncovDi,Dj→0.
Next we establish (34). Obviously (50)varn-1/2Sn′=1n∑m=1wvarηmn+2n∑1≤i<j≤wcovηin,ηjn.From (A4), (38), and Lemma 6, we have (51)1n∑1≤i<j≤wcovηin,ηjn≤1n∑1≤i<j≤ncovDi,Dj→0.On the other hand, (52)∑m=1wvar⁡ηmn=∑i=1nvar⁡Di+2∑m=1w∑km≤i<j≤km+r-1covDi,Dj-∑m=1w∑i=lmlm+q-1var⁡Di-∑i=wr+q+1nvar⁡Di.It is easy to get that (53)1n∑m=1w∑km≤i<j≤km+r-1covDi,Dj≤1n∑1≤i<j≤ncovDi,Dj→0,1n∑m=1w∑i=lmlm+q-1var⁡Di≤Cn-1wq≤Cqp→0,1n∑i=wp+q+1nvar⁡Zi≤Cn-1n-wp+q≤Cpn→0,and (54)1n∑i=1nvar⁡Zi→v0K1v12x1.Thus, (34) is verified.
As to (35), according to Lemma 4, we have (55)Eexp⁡it∑m=1wηmn-∏m=1wEexp⁡itηmn≤16wαq+1≤cnαqr,which tends to zero by (A4).
As to (36), by (A4), (46), and Lemma 6, for any ϵ1>0 and 0<2ϵ1≤λ, we have (56)1n∑m=1wEηmn2Iηmn>ϵ≤Cn-1-λ/2∑m=1wE∑i=kmkm+r-1Di2+λ≤Cn-1-λ/2∑m=1wr1+λ/2max1≤i≤n⁡EDi2+δ2+λ/2+λ+n1+ϵ1max1≤i≤n⁡EDi2+δ2+λ/2+λ≤Cn-λ/2rλ/2h1-δ2+λ/22+δ≤Crλ/2n-5λ+2λδ-δ/52+δ→0.Hence, (36) is shown. This completes the proof.Proof of Theorem3.
Note that(57)a˘1x1=1n∑i=1n∫p-1x1,x2S11x1,x2Zi1+S12x1,x2Zi2×K1,h1Xi1-x1K2,h2Xi2-x2dx2Yi1+op1,a1x1=∫a1x1q2x2dx2=∫e1,4τUτWU-1UτWUa1x1,a2x2,h1a1x1,h2a2x2τq2x2dx2=1n∑i=1n∫p-1x1,x2S11x1,x2Zi1+S12x1,x2Zi2×K1,h1Xi1-x1K2,h2Xi2-x2wix1,x2q2x2dx21+op1,where wi(x1,x2)=a1(x1)Zi1+a2(x2)Zi2+a1′(x1)(Xi1-x1)Zi1+a2′(x2)(Xi2-x2)Zi2. Then following the method of the proof of Theorem 2.1 and Theorem 2.1 in Zhang and Li [8], we can obtain this Theorem.


## 4. Conclusion Remarks

In regression analysis,α-mixing has received a lot of attentions such as Shen and Xie [17], Fan et al. [11], Chesneau et al. [18], and Guo and Liu [19]. This is because α-mixing is reasonably weak among various mixing conditions and has many practical applications. We, in this paper, focus on the varying-coefficient regression models with different smoothing variables and extend the results of Zhang and Li [8] and Zhang and Li [9] from i.i.d. assumptions to α-mixing setting. We employed the local linear technique to obtain an initial value for the estimate of each coefficient function in model (1), and, then, both averaged and integrated estimates of each coefficient function are defined by averaging and integrating its initial value on these variables which the coefficient function does not share. Under certain regularity conditions, the asymptotic properties of the proposed two estimators are established. There are some other interesting issues to be further studied; for example, the data come from survival data, such as censored data, missing data, or left truncated data. These extensions greatly expand the scope of the applicability of our methods.



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*Source: 7146793-2019-07-04.xml*