# Wavelet-M-Estimation for Time-Varying Coefficient Time Series Models

Authors: Xingcai Zhou; Fangxia Zhu
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

This paper proposes wavelet-M-estimation for time-varying coefficient time series models by using a robust-type wavelet technique, which can adapt to local features of the time-varying coefficients and does not require the smoothness of the unknown time-varying coefficient. The wavelet-M-estimation has the desired asymptotic properties and can be used to estimate conditional quantile and to robustify the usual mean regression. Under mild assumptions, the Bahadur representation and the asymptotic normality of wavelet-M-estimation are established.


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## Body

## 1. Introduction

The analysis of nonlinear and nonstationary time series, particularly with a time trend, has been very popular over the last two decades, because most time series data, coming from economics and finance data, are nonlinear or nonstationary or trending. Some nonlinear and nonstationary parametric, semiparametric, and nonparametric time series models have been proposed in the econometrics and statistics literature, for example, [1–6] and the references therein. One of the most attracted models is the time-varying coefficient time series models, which was formulated as follows:(1)Yi=XiTβti+εi,ti=in,i=1,…,n,where Yi is the response, β·=β1·,…,βp·T is a p-dimensional vector of unspecified coefficient function defined on 0,1, Xi=Xi1,…,XipT is a p-dimensional random vector, and εi is the random error.There are many smoothers proposed to estimate the time-varying coefficientβ· in models (1), and the estimator is analyzed in the large-sample theory. Robinson [4] developed the Nadaraya–Watson method and showed the consistency and asymptotic normality of local constant estimator under the assumptions that the time series Xi is a stationary α-mixing and the errors εi are i.i.d. and independent of Xi. Cai [1] proposed the local linear approach and established asymptotic properties of the proposed estimators under the α-mixing conditions and without specifying the error distribution. Hoover et al. [7] gave a smoothing spline and a locally weighted polynomial methods for longitudinal data and presented asymptotic properties. Li et al. [8] and Fan et al. [9] made statistical inference for the partially time-varying coefficient (errors-in-variables) model, respectively. These estimations are all based on local least squares, which are efficient for Gaussian errors, but least-squares estimation may perform poorly in the presence of extreme outliers. In addition, these methods are all based on an important assumption that β· is of high smoothness. In reality, the assumption may be not satisfied. In some practical areas, such as signal and image processing, objects are frequently inhomogeneous. More robust estimation methods are required.In this paper, we propose an M-type regression based on wavelet technique, which is called wavelet-M-estimation (WME), for the time-varying coefficient time series models (1). There is the considerable literature devoted to M-estimation for nonparametric regression models. Fan et al. [10] obtained asymptotic normality of the M-estimator for local linear fit under independent observations. Hong [11] established a Bahadur representation for the local polynomial estimates in nonparametric M-regression under the i.i.d. random errors. Jiang and Mack [12] and Cai and Ould-Saïd [13] considered the local polynomial M-estimator and the local linear M-estimator for dependent observations and showed some asymptotic theories of the proposed estimators. For varying coefficient models, Tang and Cheng [14] showed asymptotic normality of local M-estimators for longitudinal data, and so on. However, the above works required the smoothness of the function being estimated, for example, assuming that the function has continuous first/second derivative. With wavelets, such assumptions are relaxed considerably. Because wavelet bases can adapt to local features of curves in both time and frequency domains, wavelet provides a new technique to analyze functions with discontinuities or sharp spikes. Therefore, it is natural to have better estimators than the local kernel method in many cases. Great achievements have been made for wavelet in nonparametric models, for example, Antoniadis et al. [15]; Donoho and Johnstone [16]; Hall and Patil [17]; Härdle et al. [18]; Vidakovic [19]; Zhou and You [20]; Lu and Li [21]; and Zhou et al. [22]. To the best of our knowledge, however, M-type estimation based on wavelet technique has not been developed for the time-varying coefficient models. We use a general formulation to treat mean regression, median regression, quantile regression, and robust mean regression in one setting by WME.The article is organized as follows. Section2 describes the wavelet analysis, α-mixing sequence, and wavelet-M-estimation for time-varying coefficients. Section 3 presents the Bahadur representation and asymptotic normality of the WME under the α-mixing stationary time series sequence, and states some application of the main results. Some technical lemmas and the proofs of the main results are given in Section 4.


## 2. Wavelet-M-Estimation

As a central notion in wavelet analysis, multiresolution analysis (MRA) plays an important role for constructing a wavelet basis, which is a sequence of closed subspacesVn, n∈ℤ, in a square integrable function space ℒ2ℝ satisfying the following properties:(i)
Vm⊆Vm+1 for m∈ℤ, where ℤ denotes the integer set(ii)
∩mVm=0 and ∪mVm¯=ℒ2ℝ, where A¯ is the closure of a set A(iii)
V-spaces are self-similar, f2mx∈Vm iff fx∈V0 for m∈ℤ(iv)
There exists a scaling functionϕ∈V0 whose integer-translates span the space V0, that is, V0=f∈ℒ2ℝfx=∑kckϕx−k, and for which the set ϕ·−k,k∈ℤ is an orthonormal basis of V0By dilation and translation forϕ·, we have ϕm,kt=2m/2ϕ2mt−k, m,k∈ℤ. From (iii) and (iv), one gets ϕmk,k∈ℤ is the orthogonal bases of Vm. According to Moore–Aronszajn theorem [23], Vm is a reproducing kernel Hilbert space with a kernel(2)Kmt,s=2m∑kϕ2mt−kϕ2ms−k.For any functionf∈Vm,(3)ft=∫fsKmt,sds.Denote the kernel ofV0 as K0t,s=∑kϕt−kϕs−k, then Kmt,s=2mK2mt,2ms. For more details, we refer to Vidakovic [19].It motivates us to define a wavelet-M-estimator ofβt by(4)β^t=argminb∑i=1nρYi−XiTb∫AiKmt,sds,where ρ· is a given convex function, and Ai are intervals that partition 0,1, so that ti∈Ai. One way of defining the intervals Ai=si−1,si is by taking s0=0, sn=1, and si=ti+ti+1/2, i=1,…,n−1. As an alternative to (4), the following equation is also used to define the WME of βt, that is, to find b to satisfy(5)∑i=1nψYi−XiTbXi∫AiKmt,sds=0,where 0 is a p-dimensional zero vector. It is a natural method to obtain (5) by taking the partial derivatives of (4) with respect to b when ρ· is continuously differentiable and equating it to null, i.e., ψ·=ρ′·. In this paper, we shall apply the suitably choice function ψ· to (5), which includes many interesting cases such as least-squares estimation, the least absolute distance estimation, and quantile regression. See the monographs Huber and Ronchetti [24] and Koenker [25] for more details about the robustness of M-estimations and quantile regression, respectively.Before stating the main results, we give the definition ofα-mixing dependence, which is necessary to establish our asymptotic theory for trending time-varying coefficient time series models. Throughout, we assume that Xi,εi is a stationary α-mixing sequence. Recall that a sequence ζk,k≥1 is said to be α-mixing (or strong mixing) if the mixing coefficients,(6)αm=supPA∩B−PAPB:A∈ℱ−∞k,B∈ℱk+m∞,converge to zero as m→∞, where ℱlk denotes the σ-field generated by ζi,l≤i≤k. The notion of α-mixing is widely adopted in the study of nonparametric regression models. It is reasonably weak and is known to be fulfilled for many stochastic processes, including many familiar linear and nonlinear time series models. We refer to the monograph of Doukhan [26] and Fan and Yao [27] for some properties or more mixing conditions.


## 3. Asymptotic Theory

We first list the regularity conditions needed in the proof of the theorems although some of them might not be the weakest possible.(A1) (i) The processXi,εi is a strictly stationary α-mixing with ∑j≥1jaαjδ/2+δ<∞ for some a>δ/2+δ and δ>0. (ii) EX13+δ<∞.(A2)ρ· is a convex function, and ψ· is assumed to be any choice of the derivative of ρ·. Denote by D the set of discontinuity points of ψ·. The common distribution function F of εi satisfying FD=0.(A3)ψ· satisfies the following conditions:(i)
There exists some functionB1z such that as u→0,(7)Eψε1+uX1=z=B1zu+ou,whereB1z is continuous in a neighborhood of X with B1X≠0.(ii)
With probability 1,(8)maxEψε1+u−ψε12X1=z,Eψε1+u−ψε12+δX1=z≤B2u,holds uniformly forz in a neighborhood of X, where B2· is continuous at u=0 with B2u=Ou.(iii)
Letγz=Eψ2ε1X1=z and γz be continuous in a neighborhood of X with γz>0. Furthermore, Eψε12+δX1=z<∞.(iv)
Γx=EγX1X1X1T and Ωx=EB1X1X1X1T are nonsingular matrices.(A4) The time-varying coefficientsβj· for j=1,…,p, and the scaling function ϕ· in wavelet kernel satisfies the following conditions:(i)
βj· belongs to Sobolev space with order ν>1/2.(ii)
βj· satisfies the Lipschitz of order condition of order γ>0.(iii)
ϕ has a compact support and is in the Schwarz space with order l>ν, satisfies the Lipschitz condition with order l. Furthermore, ϕ^ξ−1=Oξ as ξ⟶0, where ϕ^ is the Fourier transform of ϕ.(A5) (i) For the designed points,maxiti−ti−1=On−1.(ii)
For some Lipschitz functionκ·,(9)ρn=maxisi−si−1−κsin=on−1.(A6) The tuning parameterm satisfies (i) n2−m→∞. (ii) n2−mn−γ+ηmlogn=o1.(iii)
Letv∗=min5/3,21+ν/3,1+2γ/3−ε1 and ε1=0 for ν≠3/2, ε1>0 for ν=3/2. Assume that n2−mv∗=O1.Some remarks on the conditions are in order.Remark 1.
Condition (A1) is the standard requirements for moments and the mixing coefficient for anα-mixing sequence. It is well known that among various mixing conditions, for example, α-, ρ-, and φ-mixing, and α-mixing is reasonably weak and can be used to depict many stochastic processes, including many familiar linear and nonlinear time series models. (A1) (i) is a very common condition, see Cai et al. [28]; Cai and Ould-Saïd [13]; and Fan and Yao [27]; among others.Remark 2.
Conditions (A2) and (A3) are often imposed to establish the large-sample theory of M-estimation in parametric or nonparametric models, see, for example, Bai et al. [29]; Cai and Ould-Saïd [13]; and Lin et al. [30]. They are mild and cover some well-known special cases, such as least-square estimation, Huber loss, and quantile. Some special examples are given as follows.Remark 3.
Conditions (A4) and (A5) are the mild regularity conditions for wavelet smoothing, which have been adopted by Antoniadis et al. [15]; Zhou and You [20]; and Zhou et al. [22]. In condition (A6), m acts as a tuning parameter, such as the bandwidth does for standard kernel smoothers; (A6) (i) and (ii) are for Bahadur representation and (A6) (i) and (iii) are for asymptotic normality, of WME. If (A6) (iii) holds, it implies (A6) (ii). There is a wide range of options to make A6 (i) and (iii) work. For example, if ν=γ=1, then υ∗=4/3. Furthermore, take 2m=On4/5, then (A6) (i) and (iii) hold.
Recall thatβ^nt=b^ based on (1). Let θ^n=n2−mb^−βt, then θ=n2−mb−βt. Set eit=XiTβti−βt. We can write the object function in equation (4) as(10)Θnθ;t=1n−12m∑i=1nρεi+eit−n−12mXiTθ∫AiKmt,sds,and denote(11)Wnt=1n−12m∑i=1nψεiXi∫AiKmt,sds.
The first theorem is crucial for establishing the asymptotic properties of the WME.Theorem 1.
Under the conditions A1–A5 (i) and (A6) (i) and (ii), for any compact subsetK∈ℝp, we have(12)supθ∈Ksupt∈0,1Θnθ;t−Θn0;t+WntTθ−12θTΩxθ=Opbn,where(13)bn=n−γ+ηm1/2+2mn3/42mn−1/2logn.If (A1)–(A5) (i) and (A6) (i) and (iii) hold, then(14)bn=2mn1/4logn.With the help of Theorem1, we can establish the Bahadur representation of WME.Theorem 2.
Under the conditions (A1)–(A5) (i) and (A6) (i) and (ii), we have(15)β^t−βt=Ωx−1∑i=1nψεiXi∫AiKmt,sds+Opdn,uniformly in t∈0,1, where(16)dn=n−γ+ηm1/4+2mn3/82mn1/4logn.If (A1)–(A5) (i) and (A6) (i) and (iii) hold, then(17)dn=2mn5/8logn.Remark 4.
Theorem2 gives the Bahadur representation of WME for time-varying coefficient time series models (1), which shows that the WME has Bahadur order is Op2m/n5/8logn. It is slightly weaker than the Bahadur order Oploglogn/nh3/4 of Hong [11], where the bandwidth h⟶0. However, Hong [11] required strong smoothness: time-varying coefficient function β· with the second-order differentiability. We have greatly relaxed this assumption. We do not need the strong assumption. Our degrees of smoothness of the function are less restrictive. See condition (A4) (i) and (ii).
With the help of Theorem2, we can establish asymptotic normality of the WME.Theorem 3.
Under the conditions (A1)–(A5) and (A6) (i) and (iii), we have(18)n2−mβ^dt−βt⟶DN0,κtω02Ωx−1ΓxΩx−1,where β^dt=β^tm with tm=⌊2mt⌋/2m, ω02=∫01E020,udu=∑k∈ℤϕ2k, and ⌊u⌋ denotes the maximum integer not greater than u.Remark 5.
To obtain an asymptotic expansion of the variance and an asymptotic normality, we need to consider an approximation toβ^t based on its values at dyadic points of order m, as Antoniadis et al. [15] have done. The main reason is that the variance of β^t as a function of t is unstable. It can be avoided by using dyadic points tm. Also see Antoniadis et al. [15] for the details. From Theorems 2 and 3, we have the uniform weak consistency of WME:(19)supt∈0,1β^t−βt=Op2mn1/2.
Next, we shall give some special cases as corollaries of Theorem3.Corollary 1.
Letρz=z2 and ψz=2z, which corresponds to the case of mean regression, which implies B1=2, γX1=Eε12X1, Γx=EγX1X1X1T, and Ωx=EX1X1T. We have(20)n2−mβ^dt−βt⟶DN0,14κtω02Ωx−1ΓxΩx−1.Corollary 2.
Letρz=zτ−Iz<0 and ψz=τ−Iz<0, which corresponds to the case of quantile regression. Assume Pε1<0X1=τ a.s. and ε1 has a continuous positive conditional density fεX· and cumulative distribution function FεX· in a neighborhood of 0 given the X1. Thus, B1=fεX0 and Γx=τ2−2τ−1FεX0EX1X1T. Let Ωx=EX1X1T. We have(21)n2−mβ^dt−βt⟶DN0,κtω02fεX−20τ2−2τ−1FεX0Ωx−1.Furthermore, ifXi and εi are mutually independent for i=1,…,n, we have(22)n2−mβ^dt−βt⟶DN0,τ1−τκtω02fε−20Ωx−1,where fε· is the density of ε1.


## 4. Technical Lemmas and Proofs

In the following sections,C is positive constant, which may be changed from line to line in the proof.Lemma 1 (see Antoniadis et al. [15] and Walter [31]).
Suppose that (A4) holds. We have(i)
E0t,s≤ck/1+t−sk and Ekt,s≤2kck/1+2kt−sk, where k is a positive integer and ck is a constant depending on k only(ii)
sup0≤t,s≤1Kmt,s=O2m(iii)
sup0≤t≤1∫01Kmt,sds≤c, where c is a positive constant(iv)
∫01Kmt,sds⟶1 uniformly in t∈0,1, as m⟶∞Lemma 2 (see Antoniadis et al. [15]).
Suppose that (A4) and (A5) (i) hold andh· satisfies (A4) (i) and (ii). Then,(23)sup0≤t≤1ht−∑i=1nhti∫AiKmt,sds=On−γ+Oηm,where(24)ηm=12mν−1/2,if12<ν<32,m2m,ifν=32,12m,ifν>32.Lemma 3 (see Lin and Lu [32]).
LetXi,i≥1 be an α-mixing sequence, X∈ℱ−∞k, and Y∈ℱk+n∞ with EXp<∞ and EYq<∞, 1/p+1/q<1. Then,(25)EXY−EXEY≤10EXp1/pEXq1/qαn1−1/p−1/q.Lemma 4 (see Pollard [33]).
Letλnθ,θ∈Θ be a sequence of random convex functions defined on a convex, open subset Θ of ℝd. Suppose λ· is a real-valued function on Θ for which λnθ→λθ in probability, for each fixed θ in Θ. Then, for each compact subset K of Θ, in probability,(26)supθ∈Kλnθ−λθ→0.For simplicity, we introduce some notations before proving. We are interested in the asymptotic behaviors ofθ, which can follow from the new optimization objective function:(27)Gnθ;t=Θnθ;t−Θn0;t.Furthermore, denote(28)Rnθ;t=Gnθ;t−EGnθ;t+WntTθ.We rewriteGnθ;t as(29)Gnθ;t=EGnθ;t−WntTθ+Rnθ;t.Following Lemmas5–7 give the asymptotic behaviors of Rnθ;t, EGnθ;t and Wnt uniformly in t∈0,1, respectively.Lemma 5.
Under the assumptions of Theorem1, for fixed θ, it holds that(30)supt∈0,1Rnθ;t=Opn2−m1/2n−γ+ηm1/2+n−12m1/4logn,where Rnθ;t is defined by (28).Proof of Lemma 5.
From the definition ofRnθ;t, we have(31)Rnθ;t=Gnθ;t+WntTθ−EGnθ;t+WntTθ≕1n−12m∑i=1nUi−EUi∫AiKmt,sds≕1n−12m∑i=1nU˜i−EU˜i,where(32)U˜i=Ui∫AiKmt,sds,Ui=ρεi+eit−n−12mXiTθ−ρεi+eit+n−12mψεiXiTθ.
By the convexity ofρ·, we have(33)Ui≤ψεi+eit−n−12mXiTθeit−n−12mXiTθ−ψεieit+n−12mψεiXiTθ=ψεi+eit−n−12mXiTθ−ψεieit−n−12mXiTθ.
Furthermore, we have(34)VarRnθ;t=n2m2Var∑i=1nU˜i≤n2m2∑i=1nEU˜i2+2∑1≤i<j≤nCovU˜i,U˜j.
Letan=2m/nn−γ+ηm+2m/n5/2. From conditions A1 (ii), A3 (ii), and A4 (ii), and Lemma 1, we have(35)∑i=1nEU˜i2=∑i=1nEEψεi+eit−n−12mXiTθ−ψεiXi2×eit−n−12mXiTθ2∫AiKmt,sds2≤∑i=1nEB2eit−n−12mXiTθeit−n−12mXiTθ2∫AiKmt,sds2≤C∑i=1nEXiTβti−βt−n−12mXiTθ3∫AiKmt,sds2≤C∑i=1nEXiTβti−βt3+n−12mXiTθ3∫AiKmt,sds2≤C∑i=1nEX13βti−βt3+n−12m3/2X13θ3∫AiKmt,sds2≤O2mnn−γ+ηm+2mn5/2=Oan.
To obtain an upper bound for the second term on the right-hand side of (34), we split it into two terms as follows. Let S1=j,i:1≤j−i≤dn;1≤i<j≤n and S2=j,i:1≤i<j≤n−S1 with dn⟶∞ specified later. We have(36)∑1≤i<j≤nCovU˜i,U˜j=∑S1CovU˜i,U˜j+∑S2CovU˜i,U˜j≕Jn1+Jn2,where dn is a sequence of positive integers such that dn=Ologn as n⟶∞.
ForJn1, by (35) and choice of dn, we have(37)Jn1≤C∑j=1dn∑i=1n−jCovU˜i,U˜i+j≤dn∑i=1nEU˜i2=dnOan=Oanlogn.
By conditions A1 (ii), A3 (ii), and A4, and Lemmas1 and 2, we have(38)EU˜i2+δ=EEψεi+eit−n−12mXiTθ−ψεiXi2+δ×eit−n−12mXiTθ2+δ∫AiKmt,sds2+δ≤CEeit−n−12mXiTθ3+δ∫AiKmt,sds2+δ≤CEX13+δβti−βt3+δ+n−12m3+δ/2X13+δθ3+δ∫AiKmt,sds2+δ≤Cβti−βt3+δ+n−12m3+δ/2∫AiKmt,sds2+δ.
ForJn2, by (38), conditions A1 (ii) and A3 (ii), and Lemmas 1–3, we have(39)Jn2≤C∑j=dn+1nαjδ/2+δ∑i=1n−dn−1EU˜i2+δ1/2+δEU˜i+dn+12+δ1/2+δ≤C∑j=dn+1nαjδ/2+δ×∑i=1n−dn−1βti−βt3+δ+n−12m3+δ/21/2+δ∫AiKmt,sds×βti+dn+1−βt3+δ+n−12m3+δ/21/2+δ∫Ai+dn+1Kmt,sds≤Cn∑j=dn+1nαjδ/2+δ×∑i=1n−dn−1βti−βt3+δ+n−12m3+δ/22/2+δ∫AiKmt,sds2+∑i=1n−dn−1βti+dn+1−βt3+δ+n−12m3+δ/22/2+δ∫Ai+dn+1Kmt,sds2≤C∑j=dn+1nαjδ/2+δ∑i=1nβti−βt3+δ+n−12m3+δ/22/2+δ∫AiKmt,sds2≤C∑j=dn+1nαjδ/2+δ∑i=1nβti−βt2+n−12m3+δ/2+δ∫AiKmt,sds2≤C∑j=dn+1nαjδ/2+δ2mnn−γ+ηm+2mn5+2δ/2+δ≤C∑j=dn+1nαjδ/2+δ2mnn−γ+ηm+2mn5/2≤Candn+1−a∑j=dn+1njaαjδ/2+δ=oan.
From (36), (37), and (39), one gets ∑1≤i<j≤nCovU˜i,U˜j=Oanlogn. Combining with (34) and (35), we have(40)VarRnθ;t=On2−m2anlogn,uniform in t∈0,1. Note that ERnθ;t=0. We have(41)Rnθ;t=Opn2−m2anlogn=Opn2−m1/2n−γ+ηm1/2+n−12m1/4logn,uniform in t∈0,1.Lemma 6.
Under the assumptions of Theorem1, for fixed θ, it holds that(42)supt∈0,1EGnθ;t−12θTΩxθ=On2−mn−γ+ηm=o1,where Gnθ;t is defined by (1).Proof of Lemma 6.
By condition A2 (i), Lemma2, and Lemma 1 in Bai et al. [29], we have(43)EGnθ;t=1n−12m∑i=1nEEρεi+eit−n−12mXiTθ−ρεi+eitXi∫AiKmt,sds=1n−12m∑i=1nE12n−12mXiTθ2−eitn−12mXiTθB1Xi1+o1∫AiKmt,sds=12θT∑i=1nEB1XiXiXiT∫AiKmt,sdsθ1+o1−1n−12mθT∑i=1nEB1XiXiXiTβti−βt∫AiKmt,sds=12θTΩxθ−n2−mOn−γ+ηm.
This finishes Proof of Lemma6.Lemma 7.
Under the assumptions of Theorem3, it holds that(44)Wndt⟶DN0,ω02κtΓx,where Wndt=Wntm with tm=⌊2mt/2m⌋, and Wnt is defined by (1).Proof of Lemma 7.
Recall that(45)Wnt=1n−12m∑i=1nψεiXi∫AiKmt,sds.
Letξi=n2−mψεiXi∫AiKmt,sds. First, we calculate its variance-covariance. Note that Eψε1X1=o1 by condition A3 (i). We have(46)VarWnt=∑i=1nEξiξiT+2∑1≤i<j≤nCovξi,ξj.
By using the same argument as those used in Proof of Lemma5, we obtain(47)∑1≤i<j≤nCovξi,ξj=o1.
Now, we prove the first term on the right-hand side of (46). Using the compact support and Lipschitz properties of ϕ, one can show that E0t,· is Lipschitz uniformly in t, so that(48)supt∈0,1E02mt,2mvi−E02mt,2mui=O2mn−1,and the Lipschitz property of κ· implies(49)κvi=κsi+On−1,where ui and vi belong to Ai. By condition A3 (iii), we have(50)∑i=1nEξiξiT=∑i=1nEEξiξiTXi=EγX1X1X1Tn2−m∑i=1n∫AiKmt,sds2=2−mnΓx∑i=1n∫AiKmt,sds2−1n∫01Em2t,sκsds+2−mΓx∫01Em2t,sκsds=n2−mΓx∑i=1nsi−si−12Km2t,ui−1nsi−si−1Em2t,viκvi+2−mΓx∫01Em2t,sκsdswhereuiandvibelong onAi.By48and49,and a standard calculation=Γxn2−mO1nOn2−mρn22m+22mn2+23mn2+2−mΓx∫01Em2t,sκsds=Oρnn+2mn−1+2−mΓx∫01Em2t,sκsds.
Since2−m∫01Em2t,sκsds is unstable as a function of t, we need to compute it at dyadic points of order m. By Lemma 6.1 in Antoniadis et al. [15], we have(51)limm→∞2−m∫01Em2tm,sκsds=κtω02.
Combining with (46), (47), and (50), we have(52)VarWndt=ω02κtΓx+o1.
Second, we shall show (44). Redefine(53)ξi=n2−mψεiXi∫AiKmt,sds−En2−mψεiXi∫AiKmt,sds.
Note thatEψε1X1=o1, and it turns (44) to(54)∑i=1nξid⟶DN0,ω02κtΓx,where the definition of ξid is similar to ξi by using tm to replace t. As ψ· is not necessarily bounded, we employ a truncation method. Denote ψL·=ψ·Iψ·≤L and as before that ξ¯id=n2−mψLεiXi∫AiKmtm,sds−En2−mψLεiXi∫AiKmtm,sds, and let Γ¯x be defined as before with γ· replaced by EψL2ε1X1=⋅. Similar to Proof of Theorem 2 in Cai et al. [28], by using Doob’s large-block and small-block technique, we can show that(55)∑i=1nξ¯id⟶DN0,ω02κtΓ¯x.
Letξ˜id=ξid−ξ¯id. To prove (54), by (55), it suffices to show(56)Var∑i=1nξ˜id⟶0,at first n⟶∞ and then L⟶∞. Let Γ˜x be defined as before with γ· replaced by Eψ2ε1Iψε1>LX1=⋅. With the same argument as (46), one gets(57)Var∑i=1nξ˜id=ω02κtΓ˜x+o1⟶0.
Since(58)Γ˜x=EEψ2ε1Iψε1>LX1X1X1T≤EEψ2+δLδε1Iψε1>LX1X1X1T≤L−δEEψ2+δX1X1X1T⟶0,asL⟶∞,from condition A3 (iii). Thus, (56) holds. Therefore, Proof of Lemma 7 is completed.Proof of Theorem 1.
From Lemmas5 and 6, and (29), for the fixed θ, we have(59)supt∈0,1Gnθ;t−12θTΩxθ+WntTθ=Opn2−mOn−γ+ηm+n2−mn−γ+ηm1/2+n−12m1/4logn=Opbn=o1.
From Lemma7, it is easy to see that Wnt is stochastically bounded. Since the convex function λnθ=Gnθ;t−Wnt converges in probability to the convex function λθ=1/2θTΩxθ, it follows from convexity Lemma 4 that for any compact set K,(60)supθ∈Ksupt∈0,1Gnθ;t−12θTΩxθ+WntTθ=Opbn=o1.
Notice that the convexity lemma strengthens the pointwise result to uniform convergence on compact subsets ofℝp. This completes Proof of Theorem 1.Proof of Theorem 2.
To obtain Bahadur representation of WME, the idea behind the proof is to approximateGnθ;t by a quadratic function whose minimizing value has an asymptotic the behavior, and then to show that θ^n lies close enough to the minimizing value to share its asymptotic behaviour. We have done the first step, that is, the results of Theorem 1 and Lemma 7. Let θ¯n=Ωx−1Wnt and cn2=bnlogn. Now, we prove the second step. The argument will be complete if we can show for each ε>0 that(61)Pθ^n−θ¯n>cnε=o1.
The argument is similar to Proof of Theorem1 in Pollard [33], whose method is extended to obtain the Bahadur representation of WME. From Theorem 1, the compact set K can be chosen to contain a closed ball Bn with center θ¯n and radius cnε, with probability arbitrarily close to one. Thereby, it implies that(62)Δn=cn−2supθ∈Bnsupt∈0,1Gnθ;t−12θTΩxθ+WntTθ=Op1logn=op1.
Now, we consider the behavior ofGnθ;t outside Bn. Suppose θ=θ¯n+cnϱυ with ϱ>ε and υ is a unit vector. Define θ∗ as the boundary point of Bn that lies on the line segment from θ¯n to θ, that is, θ∗=θ¯n+cnευ. Convexity of Gnθ;t and the definition of Δn imply(63)εϱGnθ;t+1−εϱGnθ¯;t≥Gnθ∗;t≥12θ∗TΩxθ∗+WntTθ∗−cn2Δn≥12θ¯nTΩxθ¯n+WntTθ¯n+cn212ε2υTΩxυ−Δn≥Gnθ¯n;t+cn212ε2υTΩxυ−2Δn.
Furthermore, we have(64)Gnθ;t≥Gnθ¯n;t+cn212ε2υTΩxυ−2Δn,uniformly in t∈0,1. In addition, the right-hand side of the above inequality does not depend on θ. Hence, for each ε>0 and n large enough, we have(65)Pinfθ−θ¯n>cnεGnθ;t≥Gnθ¯n;t+cn212ε2υTΩxυ−2Δn→1,which leads to θ^n−θ¯n≤cnε with probability tending to one, because 2Δn<1/2ε2υTΩxυ, which happens with probability tending to one, the minimum of Gnθ;t cannot occur at any θ with θ−θ¯n>cnε. So, (61) holds, which leads to(66)n2−mβ^t−βt=Ωx−1Wnt+Opcn,uniformly in t∈0,1. Thus, Proof of Theorem 2 is completed.Proof of Theorem 3.
From Theorem2 and Lemma 7, we can directly obtain the result of Theorem 3. Thus, we complete Proof of Theorem 3.



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*Source: 1025452-2020-09-03.xml*