# Empirical Likelihood for Partially Linear Single-Index Models under Negatively Associated Errors

Authors: Xin Qi; ZhuoXi Yu
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, the authors consider the application of the blockwise empirical likelihood method to the partially linear single-index model when the errors are negatively associated, which often exist in sequentially collected economic data. Thereafter, the blockwise empirical likelihood ratio statistic for the parameters of interest is proved to be asymptotically chi-squared. Hence, it can be directly used to construct confidence regions for the parameters of interest. A few simulation experiments are used to illustrate our proposed method.


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## Body

## 1. Introduction

The partially linear single-index model is as follows:(1)Y=gZTβ+XTθ+ε,where Y is a response variable, Z,X∈ℝp×ℝq is the covariate, g⋅ is an unknown univariate measurable function, ε is a random error, and β,θ is an unknown vector in ℝp×ℝq with β=1 (where ⋅ denotes the Euclidean norm). The restriction β=1 assures identifiability.Model (1) is flexible enough to include many important statistical models, so it has attracted much attention and has been extensively studied in recent years. Relevant studies about Model (1) have been done by [1–5]; all of which are based on the independent error sequence. In practice, all the parts of the random error sequence are often associated with each other, such as negatively associated errors, m-dependent errors, and ARCH errors, so that the abovementioned findings cannot be used directly. Therefore, it is necessary to study Model (1) with associated errors.The finite random variable sequenceξi,1≤i≤n is negatively associated (NA); if, on the condition of any two arbitrary disjoint subsets A, B⊂1,2,…,n and any real-valued coordinate-wise nondecreasing functions f1 and f2, it is established that there is(2)Covf1ξi,i∈A,f2ξj,j∈B≤0.As for infinite random variable sequence, if any arbitrary finite subset is negatively associated, the infinite sequence is negatively associated.NA sequence has been introduced and studied by the authors of [6, 7] since the 1980s. Because the NA sequence includes the independent sequence, it has been widely applied in multivariate statistical analysis, the permeability analysis, and reliability theory drew much attention, and a lot of research results have been obtained. Under the fields of NA random variables, the author of [8] presented the asymptotic normality and central limit theorem; the authors of [9] proved the law of the iterated logarithm; the authors of [10] studied the exponential inequality, and so on.However, there is little research about the partially linear single-index model under NA error. This paper, with the enlightenment of [11, 12], focuses on estimating β, θ with blockwise empirical likelihood when the errors are subjected to NA in Model (1). Throughout this paper, we assume that the data Xi,Zi,Yi,i=1,2,…,n are generated from Model (1), and εi,i=1,…,n are NA errors with Eεi|Zi,Xi=0 and Varεi|Zi,Xi=σi2<∞ in Model (1).The rest of this paper is organized as follows. In Section2, the blockwise empirical likelihood method and the relative asymptotic result are presented. In Section 3, some simulations are conducted to illustrate the proposed approach. All proofs are shown in Section 4.


## 2. Methodology and Main Results

### 2.1. Bias-Corrected Blockwise Empirical Likelihood

In this part, we will use the bias-corrected blockwise empirical likelihood to construct the confidence region forβ,θ. For this reason, first we introduce an auxiliary vector using the bias-corrected method of [2]. The details are as follows. Now that β=1 means that the value β is a boundary point on the unit sphere and gZTβ does not have a derivative at the point β of the parameter space. Nevertheless, the derivative of gZTβ on β in the building of the empirical likelihood ratio statistics must be used. Thereby, we adopt the “delete-one-component” method which is widely used in the semiparameter model. Let β=β1,β2,…,βpT and βr=β1,…,βr−1,βr+1,…,βpT be a p−1-dimensional parameter vector after deleting the rth component of β. Without losing its generality, we assume βr>0. Then, we can write(3)β=β1,…,βr−1,1−βr21/2,βr+1,…,βpT.Sinceβ can be determined by βr, only the confidence regions of βr,θ need to be taken into consideration. Moreover, βr<1, which means that β is infinitely differentiable in a neighborhood of the parameter βr. Thus, the Jacobian matrix is(4)Jβr=γ1,…,γpT,where γs1≤s≤p,s≠r is a p−1-dimensional vector with sth component 1 and γr=−1−βr2−1/2βr.Supposeμ1t=EZ|ZTβ=t and μ2t=EX|ZTβ=t and the auxiliary vector is defined as(5)ηiβr,θ=Yi−gZiTβ−XiTθ×g′ZiTβZi−μ1ZiTβTJβr,Xi−μ2ZiTβTT,where g′⋅ is the derivative of g⋅ with respect to βr. Note that Eηiβr,θ=0 on the condition that βr,θ is the true parameter. Therefore, the empirical likelihood by [13] is used to construct the bias-corrected empirical log-likelihood, which is defined as(6)lnβr,θ=−2max∑i=1nlognpi:pi≥0,∑i=1npi=1,∑i=1npiηiβr,θ=0.Sinceg⋅, g′⋅, μ1⋅, and μ2⋅ are unknown, formula (5) cannot be used to construct the confidence regions directly. To replace them with their estimators is what the researchers usually do. Next, we apply the local linear smooth method of [14] to obtain the estimators of g⋅ and g′⋅. For any fixed β,θ, we focus on finding a and b to minimize(7)∑i=1nYi−XiTθ−a−bZiTβ−t2KhZiTβ−t,where Kh⋅=K⋅/h/h, K⋅ is a kernel function and h=hn is a bandwidth. Let a^,b^ be the solution to minimize (7). Through a simple calculation,(8)g^t;β,θ=a^=∑i=1nWnit;βYi−XiTθ,(9)g^′t;β,θ=b^=∑i=1nW˜nit;βYi−XiTθ,where(10)Wnit;β=n−1KhZiTβ−tSn,2t;β−ZiTβ−tSn,1t;βSn,0t;βSn,2t;β−Sn,12t;β,W˜nit;β=n−1KhZiTβ−tZiTβ−tSn,0t;β−Sn,1t;βSn,0t;βSn,2t;β−Sn,12t;β,i=1,2,…,n,Sn,lt;β=1n∑i=1nKhZiTβ−tZiTβ−tl,l=0,1,2.Then, we define(11)μ^1t;β=∑i=1nWnit;βZi,and(12)μ^2t;β=∑i=1nWnit;βXi,as the estimators of μ1t and μ2t, respectively.When (8), (9), (11), and (12) are plugged in (5) and (6), an estimated auxiliary vector and an estimated bias-corrected empirical log-likelihood ratio can be, respectively, defined as follows:(13)η^iβr,θ=Yi−g^ZiTβ;β,θ−XiTθ×g^′ZiTβ;β,θZi−μ^1ZiTβ;βTJβr,Xi−μ^2ZiTβ;βTT,l^nβr,θ=−2max∑i=1nlognpi:pi≥0,∑i=1npi=1,∑i=1npiη^iβr,θ=0.Under the independent identically distributed errors, the empirical likelihood ratio statistic is constructed by [2]; and its asymptotic result is presented. In this paper, η^iβr,θ,i=1,…,n may be dependent when the error satisfies NA, so the method of [2] cannot be applied directly. For this, we apply the small-block and large-block arguments to construct the blockwise empirical likelihood ratio. Let k=kn=n/d+s, where a denotes the integral part of a, and d=dn and s=sn are positive integers satisfying d+s≤n. For m=1,…,k, let(14)ωn,2m−1βr,θ=enm,ωn,2mβr,θ=enm′,ωn,2k+1βr,θ=en,k+1′,where(15)enm=∑i=rmrm+d−1η^iβr,θ,enm′=∑i=lmlm+s−1η^iβr,θ,en,k+1′=∑i=kd+s+1nη^iβr,θ,rm=m−1d+s+1,lm=m−1d+s+d+1.By using the Lagrange multiplier method, the bias-corrected blockwise empirical likelihood ratio statistic is(16)l˜nβr,θ=2∑j=12k+1log1+λTβr,θωnjβr,θ,where λ∈ℝp+q−1 is determined by(17)12k+1∑j=12k+1ωnjβr,θ1+λTωnjβr,θ=0.


### 2.2. Asymptotic Result

In this subsection, the main result of this paper is summarized. In order to state the asymptotic result, the following assumptions will be used:(i)
C1: the density function ft of ZTβ is bounded away from 0 on τ and satisfies the Lipschitz condition of order 1 on τ, where τ=t=ZTβ:Z∈A and A is a compact support set of Z(ii)
C2: g⋅, μ1s⋅, and μ2l⋅ have two bounded and continuous derivatives on τ, where μ1s⋅ and μ2l⋅ are the sth and lth components of μ1⋅ and μ2⋅, 1≤s≤p,1≤l≤q, respectively(iii)
C3: the kernel K⋅ is a bounded symmetric density function and satisfies(18)∫−∞+∞u2Kudu≠0,∫−∞+∞uiKudu<∞,i=1,2,…(iv)
C4:supz,xEε4|Z=z,X=x<∞; supzEε4|Z=z<∞(v)
C5: the bandwidth h satisfies that h⟶0, nh3⟶∞, and nh8⟶∞(vi)
C6:Vβr,θ=EΛ˜Λ˜T and Σe=Covε1,ε2,…,εn are both positively definite matrices, where Λ˜=g′ZTβZ−μ1ZTβTJβr,X−μ2ZTβTT(vii)
C7:supt∈τEX2|ZTβ=t<∞(viii)
C8: when n⟶0, um=supj≥1∑i:i−j≥mCovεi,εj⟶0(ix)
C9:d=nδ1,s=nδ2,0<δ2<δ1≤1/4, and d=onh3Remark 1.
According to [2], C1–C7 guarantee the asymptotic distribution theory. C8 is a common assumption in the NA situation. C9 constrains the block size in order to obtain the desired results.Theorem 1.
Assume thatC1–C9 are satisfied. If βr,θ is the true value of the parameter and βr>0, when n⟶∞, then(19)l˜nβr,θ⟶Lχp+q−12,where ⟶L stands for the convergence in distribution.Based on Theorem1, l˜nβr,θ can be used to construct confidence regions for βr,θ. For any given 0<α<1, there exists cα which makes Pχp+q−12>cα=α tenable, and then(20)Iαβr,θ=βr,θ|l˜nβr,θ<cα,βr<1,which is the confidence regions of βr,θ with the asymptotically correct coverage probability 1−α.


## 2.1. Bias-Corrected Blockwise Empirical Likelihood

In this part, we will use the bias-corrected blockwise empirical likelihood to construct the confidence region forβ,θ. For this reason, first we introduce an auxiliary vector using the bias-corrected method of [2]. The details are as follows. Now that β=1 means that the value β is a boundary point on the unit sphere and gZTβ does not have a derivative at the point β of the parameter space. Nevertheless, the derivative of gZTβ on β in the building of the empirical likelihood ratio statistics must be used. Thereby, we adopt the “delete-one-component” method which is widely used in the semiparameter model. Let β=β1,β2,…,βpT and βr=β1,…,βr−1,βr+1,…,βpT be a p−1-dimensional parameter vector after deleting the rth component of β. Without losing its generality, we assume βr>0. Then, we can write(3)β=β1,…,βr−1,1−βr21/2,βr+1,…,βpT.Sinceβ can be determined by βr, only the confidence regions of βr,θ need to be taken into consideration. Moreover, βr<1, which means that β is infinitely differentiable in a neighborhood of the parameter βr. Thus, the Jacobian matrix is(4)Jβr=γ1,…,γpT,where γs1≤s≤p,s≠r is a p−1-dimensional vector with sth component 1 and γr=−1−βr2−1/2βr.Supposeμ1t=EZ|ZTβ=t and μ2t=EX|ZTβ=t and the auxiliary vector is defined as(5)ηiβr,θ=Yi−gZiTβ−XiTθ×g′ZiTβZi−μ1ZiTβTJβr,Xi−μ2ZiTβTT,where g′⋅ is the derivative of g⋅ with respect to βr. Note that Eηiβr,θ=0 on the condition that βr,θ is the true parameter. Therefore, the empirical likelihood by [13] is used to construct the bias-corrected empirical log-likelihood, which is defined as(6)lnβr,θ=−2max∑i=1nlognpi:pi≥0,∑i=1npi=1,∑i=1npiηiβr,θ=0.Sinceg⋅, g′⋅, μ1⋅, and μ2⋅ are unknown, formula (5) cannot be used to construct the confidence regions directly. To replace them with their estimators is what the researchers usually do. Next, we apply the local linear smooth method of [14] to obtain the estimators of g⋅ and g′⋅. For any fixed β,θ, we focus on finding a and b to minimize(7)∑i=1nYi−XiTθ−a−bZiTβ−t2KhZiTβ−t,where Kh⋅=K⋅/h/h, K⋅ is a kernel function and h=hn is a bandwidth. Let a^,b^ be the solution to minimize (7). Through a simple calculation,(8)g^t;β,θ=a^=∑i=1nWnit;βYi−XiTθ,(9)g^′t;β,θ=b^=∑i=1nW˜nit;βYi−XiTθ,where(10)Wnit;β=n−1KhZiTβ−tSn,2t;β−ZiTβ−tSn,1t;βSn,0t;βSn,2t;β−Sn,12t;β,W˜nit;β=n−1KhZiTβ−tZiTβ−tSn,0t;β−Sn,1t;βSn,0t;βSn,2t;β−Sn,12t;β,i=1,2,…,n,Sn,lt;β=1n∑i=1nKhZiTβ−tZiTβ−tl,l=0,1,2.Then, we define(11)μ^1t;β=∑i=1nWnit;βZi,and(12)μ^2t;β=∑i=1nWnit;βXi,as the estimators of μ1t and μ2t, respectively.When (8), (9), (11), and (12) are plugged in (5) and (6), an estimated auxiliary vector and an estimated bias-corrected empirical log-likelihood ratio can be, respectively, defined as follows:(13)η^iβr,θ=Yi−g^ZiTβ;β,θ−XiTθ×g^′ZiTβ;β,θZi−μ^1ZiTβ;βTJβr,Xi−μ^2ZiTβ;βTT,l^nβr,θ=−2max∑i=1nlognpi:pi≥0,∑i=1npi=1,∑i=1npiη^iβr,θ=0.Under the independent identically distributed errors, the empirical likelihood ratio statistic is constructed by [2]; and its asymptotic result is presented. In this paper, η^iβr,θ,i=1,…,n may be dependent when the error satisfies NA, so the method of [2] cannot be applied directly. For this, we apply the small-block and large-block arguments to construct the blockwise empirical likelihood ratio. Let k=kn=n/d+s, where a denotes the integral part of a, and d=dn and s=sn are positive integers satisfying d+s≤n. For m=1,…,k, let(14)ωn,2m−1βr,θ=enm,ωn,2mβr,θ=enm′,ωn,2k+1βr,θ=en,k+1′,where(15)enm=∑i=rmrm+d−1η^iβr,θ,enm′=∑i=lmlm+s−1η^iβr,θ,en,k+1′=∑i=kd+s+1nη^iβr,θ,rm=m−1d+s+1,lm=m−1d+s+d+1.By using the Lagrange multiplier method, the bias-corrected blockwise empirical likelihood ratio statistic is(16)l˜nβr,θ=2∑j=12k+1log1+λTβr,θωnjβr,θ,where λ∈ℝp+q−1 is determined by(17)12k+1∑j=12k+1ωnjβr,θ1+λTωnjβr,θ=0.


## 2.2. Asymptotic Result

In this subsection, the main result of this paper is summarized. In order to state the asymptotic result, the following assumptions will be used:(i)
C1: the density function ft of ZTβ is bounded away from 0 on τ and satisfies the Lipschitz condition of order 1 on τ, where τ=t=ZTβ:Z∈A and A is a compact support set of Z(ii)
C2: g⋅, μ1s⋅, and μ2l⋅ have two bounded and continuous derivatives on τ, where μ1s⋅ and μ2l⋅ are the sth and lth components of μ1⋅ and μ2⋅, 1≤s≤p,1≤l≤q, respectively(iii)
C3: the kernel K⋅ is a bounded symmetric density function and satisfies(18)∫−∞+∞u2Kudu≠0,∫−∞+∞uiKudu<∞,i=1,2,…(iv)
C4:supz,xEε4|Z=z,X=x<∞; supzEε4|Z=z<∞(v)
C5: the bandwidth h satisfies that h⟶0, nh3⟶∞, and nh8⟶∞(vi)
C6:Vβr,θ=EΛ˜Λ˜T and Σe=Covε1,ε2,…,εn are both positively definite matrices, where Λ˜=g′ZTβZ−μ1ZTβTJβr,X−μ2ZTβTT(vii)
C7:supt∈τEX2|ZTβ=t<∞(viii)
C8: when n⟶0, um=supj≥1∑i:i−j≥mCovεi,εj⟶0(ix)
C9:d=nδ1,s=nδ2,0<δ2<δ1≤1/4, and d=onh3Remark 1.
According to [2], C1–C7 guarantee the asymptotic distribution theory. C8 is a common assumption in the NA situation. C9 constrains the block size in order to obtain the desired results.Theorem 1.
Assume thatC1–C9 are satisfied. If βr,θ is the true value of the parameter and βr>0, when n⟶∞, then(19)l˜nβr,θ⟶Lχp+q−12,where ⟶L stands for the convergence in distribution.Based on Theorem1, l˜nβr,θ can be used to construct confidence regions for βr,θ. For any given 0<α<1, there exists cα which makes Pχp+q−12>cα=α tenable, and then(20)Iαβr,θ=βr,θ|l˜nβr,θ<cα,βr<1,which is the confidence regions of βr,θ with the asymptotically correct coverage probability 1−α.


## 3. Simulation

In this section, we use two examples to conduct some simulation studies to compare the performance of the proposed empirical likelihood method (ELM) and the normal approximation method (NAM).We assumeei∼N0.001∗i,1,i=1,…,n, and εi=ei+1−ei. Then, we can get εi∼N0.001,2 for 1≤i≤n and Covεi,εji≠j=−1 for i−j=1, otherwise Covεi,εji≠j=0. Thus, εi,1≤i≤n is a NA sequence, and its mean is approximately zero. According to C5, the rate of h is between n−1/4 and n−1/7. Therefore, the bandwidth is selected as h^=hopt by using the cross-validation (CV) method. The details of the CV method have been discussed in the references [15] of the study by Ha¨rdle et al. [15], This selected bandwidth h^ satisfies the condition C5. Let d=n1/4 and s=n1/8, which satisfy the condition C9.Example 1.
In the simulation, we generate 1000 datasets, each consisting ofn=100,200,300,500. The set of data is generated from the following model:(21)Yi=sin2π⋅ZiTβ+Xiθ+εi,1≤i≤n,where β=1/21,1T, θ=0.5, Xi∼N0,1, and Zi is the bivariate with independent U0,1 components. The kernel function is taken as the Quartic kernel Kt=15/161−t22 if t≤1.
Sinceβ1=β2, we only consider the confidence region of the parameter β1,θ. The coverage probabilities of the empirical likelihood confidence regions and the normal approximation confidence regions, with the normal level 0.95, are reported in Table 1. As is expected, the results fit our theory fairly well. The larger the sample size is, the closer the empirical coverage probability is to the nominal level. The proposed empirical likelihood method outperforms the normal approximation method.
Figure1 plots the proposed empirical likelihood confidence region and the normal approximation confidence regions for β1,θ based on the confidence level of 0.95 when the sample size is 300.Table 1
Simulation results of Example1. The coverage probabilities of the confidence regions for β1,θ when the nominal level is 0.95, respectively.
Methodn=100n=200n=300n=500ELM0.98090.97140.97020.9651NAM0.91260.92090.92150.9257Figure 1
Simulation results of Example1. Confidence region for β1,θ of Model (21) with the confidence level of 0.95.Example 2.
In this simulation, the coverage probabilities and average lengths of confidence intervals are calculated by the proposed empirical likelihood method and the normal approximation method. Consider Model (1) with p=3 and q=2, where β=1/141,2,3T, θ=0.2,0.7T, and g⋅=1.5sin⋅. Z is independent and all from the uniform U0,1, and the two components of X are from the bivariate standard normal distribution. The kernel function is taken as the Epanechnikov kernel Kt=3/41−t22 if t≤1.
Based on 500 simulation runs, the simulation results are reported in Table2. From Table 2, the following results can be obtained. The coverage probabilities of the empirical likelihood method and the normal approximation method are in agreement with the nominal level of 0.90; the empirical likelihood method has slightly smaller interval lengths compared with the normal approximation method.Table 2
Simulation results of Example2. The average lengths (AI) and empirical coverage probabilities (CP) of the confidence intervals for β1, β2, β3, θ1, and θ2 with sample size n = 200 when the nominal level is 0.90.
Methodβ1β2β3θ1θ2ELMAI0.10930.07610.23160.33220.2895CP0.93270.93110.95250.95230.9412NAMAI0.16520.10530.31870.48940.3527CP0.86460.87180.84610.84410.8609


## 4. Proofs

In order to prove Theorem1, we first give some lemmas. Throughout this section, for a concise and convenient representation, we use c0<c<∞ to denote any constant which may take a different values for each appearance, use λminA and λmaxA to denote the smallest and largest eigenvalues of A, respectively, and write A⊗2=A⋅AT.Lemma 1.
Letai,i=1,2,…,n and bi,i=1,2,…,n be any two sequences, then(22)∑i=1naibi≤c⋅max1≤i≤nbi⋅max1≤k≤n∑i=1kaji,where j1,j2,…,jn is an arbitrary arrangement of order n.The proof of Lemma1. Firstly, we assume bi≥0,i=1,2,…,n. Via Abel inequality, it follows that(23)∑i=1naibi≤2max1≤i≤nbi⋅max1≤k≤n∑i=1kaji.Secondly, we assumebi<0,i=1,2,…,n, and it also follows that(24)∑i=1naibi=∑i=1n−ai⋅−bi≤2max1≤i≤nbi⋅max1≤k≤n∑i=1kaji.Combining (23) and (24), then we get(25)∑i=1naibi≤∑i:bi≥0aibi+∑i:bi<0−ai−bi≤c⋅max1≤i≤nbi⋅max1≤k≤n∑i=1kaji.Lemma 2.
Leta1,a2,…,an be any random variables with max1≤i≤nE|ai|s≤c<∞ for some constants s>0andc>0. Then,(26)max1≤i≤nai=Opn1/s.Refer to the proof of Lemma1 of [11].Lemma 3.
Letξj:1≤j≤n be NA random variables with Eξj=0,Eξjs<∞s>1 and aj:j≥1 be a sequence of real constants. Then, there exists a constant c>0 which only depends on the given number s so that(27)E∑j=1najξjs≤c∑j=1nEajξjs,for1<s≤2,E∑j=1najξjs≤c∑j=1nEajξjs+∑j=1nEajξj2s/2,fors>2.The proof of Lemma3 can be finished with the work by [16].Lemma 4.
Assume thatC1–C4 hold, for any integer 2≤r≤4, we can obtain, uniformly over 1≤i≤n,(28)Eφ^ZiTβ;β,θ−φZiTβr=Oh2r+On−r/2h1−r,Eg^′ZiTβ;β,θ−g′ZiTβr=Ohr+On−r/2h1−2r,where φ^⋅=g^⋅,μ^1s⋅,μ^2l⋅ and φ⋅=g⋅,μ1s⋅,μ2l⋅. Here, μ1s⋅andμ2l⋅ are, respectively, the sth and lth component of μ1⋅andμ2⋅, 1≤s≤p,q≤l≤q.The proof of Lemma4 is similar to the proof of Lemma 3 by [12]. So, the details are omitted here.Lemma 5.
Suppose that it satisfiesC1–C9, βr,θ is the true value of the parameter, and the rth component of β>0, then(29)1n∑i=1nη^iβr,θ⟶LN0,Γ,and(30)1n∑i=12k+1ωniωniT−Γ=op1,where Γ=EΛ˜ε⊗2, and Λ˜ is defined in C6.The proof of Lemma5. It is easy to show that(31)1n∑i=1nη^iβr,θ=1n∑i=1nΛ˜iεi+∑v=14Mv,where(32)M1=M11TJβr,M12TT,M2=M21TJβr,O1×qT,M3=M31TJβr,M32TT,M4=M41TJβr,O1×qT,M11=1n∑i=1nμ1ZiTβ−μ^1ZiTβ;βg′ZiTβεi,M12=1n∑i=1nμ2ZiTβ−μ^2ZiTβ;βεi,M21=1n∑i=1ng^′ZiTβ;β,θ−g′ZiTβ⋅Zi−μ^1ZiTβ;βεi,M31=1n∑i=1ngZiTβ−g^ZiTβ;β,θ⋅Zi−μ^1ZiTβ;βg′ZiTβ,M32=1n∑i=1ngZiTβ−g^ZiTβ;β,θ⋅Xi−μ^2ZiTβ;β,M41=1n∑i=1ngZiTβ−g^ZiTβ;β,θ⋅g^′ZiTβ;β,θ−g′ZiTβ⋅Zi−μ^1ZiTβ;β.In order to prove (29), we firstly need to show that(33)1n∑i=1nΛ˜iεi⟶LN0,Γ.LetΔn=1/n∑i=1nΛ˜iεi2Λ˜iT. For the central limit theorem of nΔn−1/2∑i=1nΛ˜iεi, we refer to Lemma 5 of [11]; then, we can obtain (33).Secondly, we need to show thatMu1⟶P0u=1,2,3,4 and Mm2⟶P0m=1,3. Here, we consider M12. Let M12,s denote the sth component of M12. By C2 and C4 and Lemma 1 and Lemma 4, we can get(34)EM12,s2≤c⋅supzEεi2|Zi=z∑i=1nEμ2sZiTβ−μ^2sZiTβ;β2≤ch4+cnh−1⟶0.The other formulas mentioned above can be proved by using Lemma1, Lemma 3, and Lemma 4 and the similar methods of [2]. These formulas, together with (31) and (33), complete the proof of (29).The proof of (30) uses a similar method to the proof of Lemma 5 by [12]. Here, we only give some key steps. For any c∈ℝp+q−1 with c=1, we will show that(35)1ncTΓ−1/2∑i=12k+1ωniωniTΓ−1/2c=1+op1.Then, we need to show that(36)1ncTTn1c=1+op1,(37)1ncTTn2c=op1,(38)1ncTTn3c=op1,where Tn1=Γ−1/2∑i=1kenieniTΓ−1/2,Tn2=Γ−1/2∑i=1keni′eni′TΓ−1/2,andTn3=Γ−1/2en,k+1′en,k+1′T,Γ−1/2.Similar to (31), we have(39)1ncTTn1c=1n∑i=1kcTΓ−1/2∑j=riri+d−1Λ˜jεj+cTΓ−1/2∑j=riri+d−1Rnj2≕1n∑i=1kvi1+vi22,where(40)Rnj=εjg′ZjTβμ˜1jTJβr,εjμ˜2jTT−εjg˜j′Z˜jTJβr,0T+g˜jg′ZjTβZ˜jTJβr,g˜jX˜jTT−g˜jg˜j′Z˜jTJβr,0T,μ˜1j=μ1ZjTβ−μ^1ZjTβ;β,μ˜2j=μ2ZjTβ−μ^2ZjTβ;β,Z˜j=Zj−μ^1ZjTβ;β,X˜j=Xj−μ^2ZjTβ;β,g˜j=gZjTβ−g^ZjTβ;β,θ,g˜j′=g′ZjTβ−g^′ZjTβ;β,θ.As arguments in [11], we can get (36). Similar decomposition and proof can be used in (37) and (38). Thus, (30) is completely proved.Lemma 6.
UnderC1–C9, it follows that(41)max1≤i≤2k+1ωni=opn1/2.The proof of Lemma6. Now that(42)max1≤i≤2k+1ωni≤c⋅max1≤i≤nη^iβr,θ,we only need to prove(43)max1≤i≤nη^iβr,θ=opn1/2.Letη^i,s denote the sth component of η^i, and the notation of Lemma 5 is used.We consider the laterq components of η^iβr,θ. By a simple calculation, we can obtain(44)max1≤i≤nη^i,sβr,θ≤c⋅max1≤i≤nΛ˜i,sεi+c⋅max1≤i≤nμ˜2i,sεi+c⋅max1≤i≤ng˜iX˜i,s≕An1,s+An2,s+An3,s,p≤s≤p+q−1.Applying Lemma2 with Eεi2<∞,(45)max1≤i≤nεi=Opn1/2.This implies thatAn1,s=opn1/2 since EΛ˜i,sεi2<∞. Combining the Markov inequality, Lemma 4 and C4, for any δ>0,(46)PAn2,s>δn≤∑i=1nPμ2sZiTβ−μ^2sZiTβ;βεi>δn,≤1nδ2supzEεi2|Zi=z∑i=1nEμ2sZiTβ−μ^2sZiTβ;β2,≤ch4+cnh−1⟶0.Hence,An2,s=opn1/2. An3,s can be dealt similarly. Then,(47)max1≤i≤nη^i,sβr,θ=opn1/2,forp≤s≤p+q−1.Moreover,(48)max1≤i≤nη^i,sβr,θ=opn1/2,for1≤s≤p−1,can be similarly dealt with. Thus, (43) can be shown.Lemma 7.
Under conditionsC1–C9, it follows that(49)λβr,θ=Opn−1/2.The proof of Lemma7. Let ω¯=1/2k+1∑j=12k+1ωnj and S1=1/2k+1∑j=12k+1ωnjωnjT. Denoting ρ=λ, then λ can be expressed by λ=ρς, where ς∈ℝp+q−1 and ς=1.From (17), we have(50)0=12k+1ςT∑j=12k+1ωnj1+λTωnj=12k+1ςT∑j=12k+1ωnj−ρ2k+1∑j=12k+1ςTωnj21+ρςTωnj,≤12k+1ςT∑j=12k+1ωnj−ρ1+ρmax1≤j≤2k+1wnj12k+1∑j=12k+1ςTωnj2,≤12k+1ςT∑j=12k+1ωnj−ρ1+ρmax1≤j≤2k+1wnjλminS1.It follows that(51)ςTω¯≥ρλminS1−ρςTω¯max1≤j≤2k+1wnj.By a straightforward calculation, we can get(52)ςTλmaxΓ1/2Γ−1/2ω¯≥ρλminΓ1/2S1Γ−1/2λminΓ−ρςTλmaxΓ−1/2Γ1/2ω¯max1≤j≤2k+1wnj.As discussed in [12], we can obtain(53)Γ=Opn.Using Lemma5, Lemma 6, and (53), we can have(54)ρ1−max1≤j≤2k+1wnjOpn−1/2≤Opn−1/2.Therefore, it is easy to obtain(55)ρ=λβr,θ=Opn−1/2.The proof of Theorem1. Let γj=λTωnj, and the notation of Lemma 7 is used. Using a Taylor expansion in (16), we have(56)l˜nβr,θ=2∑j=12k+1log1+γj=2∑j=12k+1γj−∑j=12k+1γj2+ϕ1=22k+1λTω¯−2k+1λTS1λ+ϕ1,where(57)ϕ1≤∑j=12k+1γj3≤λT32k+1max1≤j≤2k+1wnj12k+1∑j=12k+1ωnjωnjT=op1.Thus,(58)l˜nβr,θ=22k+1λTω¯−2k+1λTS1λ+op1.By calculating directly from (17), we obtain(59)0=2k+1Γ−1/2ω¯−2k+1Γ−1/2S1Γ−1/2Γ−1/2λ+ϕ2,where(60)ϕ2=∑j=12k+1Γ−1/2ωnjγj21+γj≤λ2Γ−1/2max1≤j≤2k+1wnj∑j=12k+1ωnjωnjT1+λTωnj=opn−1/2.Therefore, it follows that(61)2k+1Γ−1/2S1Γ−1/2Γ−1/2λ=2k+1Γ−1/2ω¯+opn−1/2.It is easy to get(62)λ=Γ−1ω¯Γ−1/2S1Γ−1/2+opn−1/2.Combining (61) and (62), (58) can be rewritten as(63)l˜nβr,θ=2k+1Γ−1/2ω¯TΓ−1/2ω¯Γ−1/2S1Γ−1/2+op1=2k+1Γ−1/2ω¯T2k+1Γ−1/2S1Γ−1/2−12k+1Γ−1/2ω¯+op1.Applying Lemma5, we have(64)2k+1Γ−1/2ω¯T2k+1Γ−1/2S1Γ−1/2−12k+1Γ−1/2ω¯⟶Lχp+q−12.Thus, we complete the proof of Theorem1.



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