# Block Empirical Likelihood for Semiparametric Varying-Coefficient Partially Linear Errors-in-Variables Models with Longitudinal Data

Authors: Yafeng Xia; Hu Da
Journal: Journal of Probability and Statistics
 (2013)

---

## Abstract

Block empirical likelihood inference for semiparametric varying-coeffcient partially linear errors-in-variables models with longitudinal data is investigated. We apply the block empirical likelihood procedure to accommodate the within-group correlation of the longitudinal data. The block empirical log-likelihood ratio statistic for the parametric component is suggested. And the nonparametric version of Wilk’s theorem is derived under mild conditions. Simulations are carried out to access the performance of the proposed procedure.


---

## Body

## 1. Introduction

For longitudinal data, we consider semiparametric varying-coefficient partially linear model which has the following form:(1)Y(t)=X(t)τβ+Z(t)τθ(t)+ϵ(t),
where Y(t) is the response variable, X, Z, and t are regressors, β=(β1,…,βp)τ is a p-dimensional vector of unknown parameters, θ(t)=(θ1(t),…,θq(t))τ is a q-dimensional vector of smooth functions of time t, and ϵ(t) is a zero-mean stochastic process. Due to the curse of dimensionality, for simplicity, we assume that t is univariate.Obviously, model (1) contains many usual parametric, nonparametric, and semiparametric models. Model (1) has been studied by many authors. Zhang et al. [1] suggested a two-step method for estimating it. Li et al. [2] suggested a local least-squares procedure with a kernel weight function. Fan and Huang [3] developed a profile least-squares technique for estimating parametric model. You and Zhou [4] and Huang and Zhang [5] suggested the estimator of the parametric and nonparametric models, respectively. Fan et al. [6] proposed a semiparametric estimation of the working correlation matrix and applied a profile weighted least-squares approach.However, in many practical situations, these variables are often measured with error. In this paper, we consider this case where the variableX(t) is measured with additive error and both Z(t) and t are measured exactly. That is, X(t) cannot be observed, but an unbiased measure of X(t), denoted by W(t), can be obtained as follows:(2)W(t)=X(t)+U(t),
where U(t) is the measurement error, which is independent of (Xτ(t), Zτ(t), ϵ(t), t), with mean zero and covariance matrix Σuu. We can assume that Σuu is known. If Σuu is unknown, we estimate it by repeatedly measuring W(t) by Liang et al. [7]. For errors-in-variables models (1) and (2), Liang et al. [8] developed a profile least-squares procedure to estimate the parametric component and derived the asymptotic normality of the resulting estimator.The empirical likelihood, which is a nonparametric approach for constructing confidence regions, was introduced by Owen [9] and has many nice statistical properties (see Owen [10]). Owen [11] applied empirical likelihood to linear regression models and Kolaczyk [12] made further extensions to generalized linear models. Recently, Xue and Zhu [13] considered the varying coefficient models. You and Zhou [4], Huang and Zhang [5], and Zhao and Xue [14] investigated the empirical likelihood confidence regions for varying-coefficient partially linear models. Other related papers contain Yang and Li [15], Hu et al. [16], Wang et al. [17], and Fan et al. [18, 19].In this paper, we consider models (1) and (2) with longitudinal data; one aim of this paper is to construct the confidence region for the parameter components. To achieve it, we apply the block empirical likelihood approach [20] to construct block empirical log-likelihood ratio statistic for parameter β and then prove nonparametric Wilk’s phenomenon. Simulation studies assess the proposed method. The other aims are to prove that the maximum empirical likelihood estimator (MELE) for the parameter is asymptotically normal under some suitable conditions.The rest of this paper is organized as follows. In Section2, we construct the block empirical likelihood based confidence region for the parametric components. Assumption conditions and main results are given in Section 3. Simulation results are reported in Section 4. The proofs of the main results are stated in Section 5. Finally, some concluding remarks are given.


## 2. Methodology

In this section, we are to extend the result of Hu [21] to the semivarying coefficient errors-in-variables model with longitudinal data.We apply longitudinal data (Yi(tij), Xi(tij), Zi(tij), tij). i=1,…,n, and j=1,…,ni which are generated from semivarying coefficient errors-in-variables model through the following equation:
(3)Yij=Xijτβ+Zijτθ(tij)+ϵij,Wij=Xij+Uij,
where Yij=Yi(tij), Xij=Xi(tij), Zij=Zi(tij) and ϵij=ϵi(tij), and Uij=Ui(tij), i=1,…,n, j=1,…,ni. We use counting process Ni(t)≡∑j=1ni‍I(tij≤t) to describe the number of observations of the ith subject. We assume that ni is bounded, but the number of subjects n goes to infinity.Suppose thatβ is known; then, model (3) can be reduced to a varying-coefficient regression model:
(4)Yij-Xijτβ=∑k=1q‍Zijkθk(tij)+ϵij,hhhhi=1,…,n,j=1,…,ni.
Here, the local linear regression method is applied to estimate the coefficient function {θj(·),j=1,…,q} in model (4). That is, for t in a small neighborhood of t0, one can approximate θj(·) locally by a linear function
(5)θj(t)≈θj(t0)+θj′(t0)(t-t0)≡aj+bj(t-t0),j=1,…,q,
where θj′(t)=∂θj(t)/∂t. This leads to the following weighted least-squares problem: find {(aj,bj),j=1,…,q} to minimize
(6)∑i=1n‍∫01‍{Yi(s)-Xiτ(s)β-∑k=1q‍{ak+bk(s-t)}Zik(s)}2×Kh(s-t)dNi(s),
where K is a kernel function, Kh(·)=K(·/h)/h, and h is a bandwidth. Let
(7)Ωt=diag(Kh(t11-t),…,Kh(t1n1-t),…,Kh(tnnn-t))Y=(Y1(t11),…,Y1(tn1),…,Yn(tnn))τ,X=(X1τ,…,Xnτ)τ,Z=(Z11τ⋮Z1n1τ⋮Znnnτ)=(Z1(t11)⋯Z1(t1q)⋮⋱⋮Z1(tn11)⋯Z1(tn1q)⋮⋱⋮Zn(tnn1)⋯Zn(tnnq)),hhhϵ=(ϵ11,…,ϵ1n1,…,ϵnnn)τ,(8)Dt=(Z11⋯Z1n1⋯Znnn(t11-t)hZ11⋯(t1n1-t)hZ1n1⋯(tnnn-t)hZnnn)τ.Then, the solution to problem (6) is given by
(9)(a^1(t),…,a^q(t),hb^1(t),…,hb^q(t))τ=(DtτΩtDTτ)-1DtτΩt(Y-Xτβ).
Then, θ^(t) can be given by
(10)θ^(t)=(Iq,0q)(DtτΩtDTτ)-1DtτΩt(Y-Xτβ),
where Iq is q×q identity matrix and 0q is q×q zero matrix. Denote
(11)(Iq,0q)(DtτΩtDTτ)-1DtτΩt≡(S11(t),…,Snnn(t));
then,
(12)θ^(tij)=∑l=1n‍∑m=1nl‍Slm(tij)(Ylm-Xlmτβ).
Substituting (12) into (4), we can obtain the approximate residuals as the following:
(13)r^ij(β)=Yij-Xijτβ-Zijτ∑l=1n‍∑m=1nl‍Slm(tij)(Ylm-Xlmτβ)=Y~ij-X~ijτβ,
where
(14)Y~ij=Yij-Zijτ∑l=1n‍∑m=1nl‍Slm(tij)Ylm,X~ij=Xij-(∑l=1n‍∑m=1nl‍Slm(tij)Xlm)τZij.Similar to Owen [10], {r^ij(β),i=1,…,n;j=1,…,ni} can be treated as a random sieve approximation of the random error sequence {ϵij,i=1,…,n;j=1,…,ni}. In order to deal with the correlation within group, we use the block empirical likelihood method. The block empirical likelihood procedure takes the ‘‘data” r^ij(β),j=1,…,ni into account as a whole. Hence, similar to Xue and Zhu [13], we introduce the auxiliary random vector
(15)η~i(β)=∫01‍X~i(t)[Y~i(t)-X~iτ(t)β]dNi(t).Following (13), if β is true, then E{η~i(β)}=o(1). If one ignores the measurement error and replaces Xij by Wij in η~i(β), one can show that the resulting estimator is inconsistent. As we all know, inconsistency caused by the measurement error can be overcome by applying the so-called correction for attenuation proposed by Fuller [22] in linear regression. With a similar way as in Zhao and Xue [14], the corrected-attenuation auxiliary vector is introduced and defined as
(16)ηˇi(β)=∫01‍{W~i(t)(Y~i(t)-W~iτ(t)β)+Σuuβ}dNi(t),
where W~ij=Wij-(∑l=1n‍∑m=1nl‍Slm(tij)Wlm)τZij. The term Σuuβ aims to avoid the underestimating for the parameter caused by the measurement error. Therefore, the empirical likelihood ratio function for β is defined as
(17)ℛ(β)=max{∏i=1n‍(npi)∣pi≥0,∑i=1n‍pi=1,∑i=1n‍piηˇi(β)=0}.
A unique value for ℛ(β) exists, provided that 0 is inside the convex hull of the point (ηˇ1(β),…,ηˇn(β)). Using the Lagrange multiplier technique, the optimal value for pi is
(18)pi=1n{1+λτηˇi(β)}-1,i=1,…,n,
where λ=(λ1,…,λn)τ is the solution of the equation
(19)1n∑i=1n‍ηˇi(β)1+λτηˇi(β)=0.
Then, the block empirical log-likelihood ratio function is
(20)ℒℛ(β)=-2logℛ(β)=2∑i=1n‍log(1+λτηˇi(β)).
In addition, by maximizing ℒℛ(β), we can obtain the maximum empirical likelihood estimator (MELE) βˇ. Let
(21)Γˇ=1n∑i=1n‍∫01‍(W~i(t)W~iτ(t)-Σuu)dNi(t).
If the matrix Γˇ is invertible, then the MELE of β can be given by
(22)βˇ=Γˇ-11n∑i=1n‍∫01‍W~i(t)Y~i(t)+op(n-1/2).
According to βˇ, we can define the estimator {θj(·),j=1,…,q} as
(23)θˇ(t)=(θˇ1(t),…,θˇq(t))τ=(Iq,0q)(DtτΩtDTτ)-1DtτΩt(Y-Xτβˇ).


## 3. Main Results

To establish asymptotic properties of the block empirical log-likelihood ratio, we make the following assumptions. These assumptions are made by You and Zhou [4]. We use ∥·∥ to denote the Euclidean norm with ∥a∥=(a12+⋯+an2)1/2 and a=(a1,…,an)τ.Assumption 1.
The random variablet has a compact support Ξ. The density function f(·) of t has a continuous second derivative and is uniformly bounded away from zero.Assumption 2.
Thep×p matrix E(XXτ∣t) is nonsingular for each t∈Ξ. E(XXτ∣t), E(XXτ∣t)-1, and E(XZτ∣t) are all Lipschitz continuous.Assumption 3.
There is as>2 such that E∥X∥2s<∞, E∥Z∥2s<∞, E∥ϵ∥2s<∞, and E∥t∥2s<∞ and for some ϵ<2-s-1 such that n2ϵ-1h→∞ as n→∞.Assumption 4.
α
j
(
·
)
,
j
=
1
,
…
,
q have the continuous second derivative in t∈Ξ.Assumption 5.
The kernelK(·) is a symmetric probability density function and is a bounded variation function on its support.Assumption 6.
The bandwidthh satisfies nh8→0 and nh2/(logn)2→∞ as n→∞.The following theorem gives the asymptotic distribution ofℒℛ(β).Theorem 1.
Assume that the Assumptions1–6 hold; if β is the true value of the parameter, then
(24)ℒℛ(β)⟶𝒟χp2asn⟶∞,
where →𝒟 denotes the convergence in distribution and χp2 is a chi-square distribution with p degrees of freedom.Then, we can construct the confidence regions for the parameterβ. More precisely, for any 0<α<1, let Cα be such that p(χp2>Cα)≤1-α. Then,
(25)ℋ(α)={β∈Rp:ℒℛ(β)≤Cα}
constitute a confidence region for β with asymptotic coverage 1-α.Theorem 2.
Assume that the Assumptions1–6 hold. Then, one has
(26)n(βˇ-β)⟶𝒟N(0,Γ-1ΣΓ-1),
where
(27)Γ=E{∫01‍[W(t)-μtτZ(t)]⊗2dN(t)},B=E{∫01‍[W(t)-μtτZ(t)]ɛ(t)}⊗2,A⊗2=AAτ,μ(t)=E{Zτ(t)W(t)∣t}-1E{Z(t)Zτ(t)∣t}.


## 4. Simulation Results

In this section, we will conduct some simulations to the empirical likelihood (EL) method. The data are generated from(28)yi(tij)=xi(tij)β+z1i(tij)θ1(tij)+z1i(tij)θ2(tij)+ϵi(tij),wi(tij)=xi(tij)+ui(tij),
where wi(t)~N(0,1), β=1.5, z1i(t)~N(0,1), z2i(t)~N(0,1), t~U(0,1), θ1(t)=sin(2πt), θ2(t)=cos(2πt), ϵi(t)~N(0,1), ui(t)=bei(tj)+ei(tj-1), and ei(t)~N(0,1).In the simulation studies, for each combination ofni, and b, we draw 1,000 random samples of sizes 100 or 200 from the above model, respectively. For each sample, a 95% confidence interval for β=1.5 is computed using our block empirical likelihood method. The kernel function is taken as the Gauss kernel Kh(t)=(1/2πh)exp(-(t)2/2h2). The “leave-one-sample-out” method is used to select the bandwidth h. We define the score of h as follows:(29)CV(h)=1n∑i=1n‍∫01‍{Yij-Wijτβˇ-i-Zijτθˇ-i(tij)}2hhhhhhh-βˇ-iτΣvvβˇ-idNi(t).
Then cross-validation smoothing parameter hCV is the minimizer of CV(h). Some representative coverage probabilities are reported in Table 1.Table 1
Coverage probabilities (CP) and average lengths (AL) of the confidence intervals forβ=1.5 and σ2=0.2.
k
Number of replicates
CP (%)
AL
NA
EL
NA
EL
b
=
0.3
100
n
1
=
⋯
=
n
50
=
3
93.27
93.34
0.2877
0.2789
n
51
=
⋯
=
n
100
=
3
100
n
1
=
⋯
=
n
50
=
3
91.94
92.16
0.3079
0.3005
n
51
=
⋯
=
n
100
=
2
200
n
1
=
⋯
=
n
100
=
3
93.98
94.26
0.2439
0.2432
n
101
=
⋯
=
n
200
=
3
200
n
1
=
⋯
=
n
100
=
3
93.61
94.74
0.2651
0.2176
n
101
=
⋯
=
n
200
=
2
b
=
0.6
100
n
1
=
⋯
=
n
50
=
3
93.05
93.21
0.2967
0.2936
n
51
=
⋯
=
n
100
=
3
100
n
1
=
⋯
=
n
50
=
3
91.74
91.86
0.3175
0.3114
n
51
=
⋯
=
n
100
=
2
200
n
1
=
⋯
=
n
100
=
3
93.47
94.02
0.2711
0.2437
n
101
=
⋯
=
n
200
=
3
200
n
1
=
⋯
=
n
100
=
3
92.82
93.69
0.2981
0.2646
n
101
=
⋯
=
n
200
=
2


## 5. Proof of the Main Results

In order to prove the main results, we first introduce several lemmas. Letuk=∫‍tkK(t)dt, vk=∫‍tkK2(t)dt, k=0,1,2,4, cn=h2+((logn/n)h)1/2, G(T)=E(ZZτ∣T), Ψ(T)=E(XZτ∣T), and M=(Z1τθ(t1),…,Znτθ(tn))τ.Lemma 3.
Let(X1,Y1),…,(Xn,Yn) be i.i.d random vector, where Yi is scalar random variable. Further, assume that E|Y1|s<∞, supx∫‍|y|sf(x,y)dy<∞, where f(·,·) denotes the joint density of (X,Y). Let K(·) be a bounded positive function with a bounded support, satisfying a Lipschitz condition. Given that n2ϵ-1h→∞ for some ϵ<1-s-1, then size
(30)supx|1n∑i=1n‍{Kh(Xi-x)Yi-E[Kh(Xi-x)Yi]}|=Op({log(1/h)nh}1/2).Proof.
This lemma can be found in Mack and Silverman [23].Lemma 4.
Letϵi,i=1,…,n, be a sequence of multi-independent random variate with E(ϵi)=0 and E(ϵi2)<c<∞. Then,
(31)max1≤k≤n|∑i=1k‍ϵi|=Op(nlogn).
Further, let (j1,…,jn) be a permutation of (1,…,n). Then, one has
(32)max1≤k≤n|∑i=1k‍ϵji|=Op(nlogn).Proof.
We can prove this lemma immediately by Kolmogorov inequality.Lemma 5.
LetD1,…,Dn be i.i.d random variables. If E|Di|s are uniformly bounded for s>1, then one has
(33)max1≤i≤n|Di|=o(n1/s).Proof.
This lemma can be found in Shi and Lau [24].Lemma 6.
Suppose that Assumptions1–6 hold; one has
(34)DtτΩtDt=Nf(t)Γ⊗G(t)(1+Op(cn)),DtτΩtXt=Nf(t)(1,0)τ⊗Ψ(t)(1+Op(cn)),DtτΩtZt=Nf(t)(1,0)τ⊗G(t)(1+Op(cn)),
which hold for all t∈[a,b]⊂[0,1], where Γ=diag(1,u2).Proof.
This follows immediately from the result that was obtained by Yang and Li [15].Lemma 7.
Suppose that Assumptions1–6 hold; one has, when tij∈[a,b],
(35)Siτ(tij)X=Ziτ(tij)G-1(tij)Ψ(tij)(1+Op(cn)),Siτ(tij)M=Ziτ(tij)θ(tij)(1+Op(cn)).Proof.
LetSiτ(tij)=[Ziτ(tij),0](DtijτΩtijDtij)-1DtijτΩtij; then, Lemma 7 can be directly attained by Lemma 6.Lemma 8.
Suppose that the Assumptions1–6 hold, one has
(36)1n∑i=1n‍ηˇi(β)⟶𝒟N(0,Σ),1n∑i=1n‍ηˇi(β)ηˇiτ(β)⟶𝒫Σ,max1≤x≤n∥ηˇi(β)∥=Op(n1/2),
where Σ is defined by (26).Proof of Theorem1.
From (36), using the same arguments as were used in the proof of Owen [10], we have
(37)∥λ∥=Op(n-1/2),
where λ is defined in (19). Then, we have size
(38)0=1n∑i=1n‍ηˇ(β)1+λτηˇi(β)=1n∑i=1n‍ηˇi(β)-1n∑i=1n‍ηˇi(β)ηˇiτ(β)λ+1n∑i=1n‍ηˇi(β)(λτηˇi(β))21+λτηˇi(β).
By using Lemma 8, we obtain
(39)∑i=1n‍(λτηˇi(β))2=∑i=1n‍λτηˇi(β)+Op(1),λ=[∑i=1n‍ηˇi(β)ηˇiτ(β)]-1∑i=1n‍ηˇi(β)+op(n-1/2).
Applying the Taylor expansion to (20), we get that
(40)ℒℛ(β)=2∑i=1n‍[λτηˇi(β)-12(λτηˇi(β))2]+op(1).
Hence, together with (39), we have size
(41)ℒℛ(β)=[1n∑i=1n‍ηˇi(β)]τ×[1n∑i=1n‍ηˇi(β)ηˇiτ(β)]-1[1n∑i=1n‍ηˇi(β)]+op(1).
Together with Lemma8, this proves Theorem 1.Proof of Theorem2.
Following the similar arguments as were used in the proof of Theorem 2 in Yang and Li [15], we have
(42)βˇ-β=Γˇ-11n∑i=1n‍ηˇi(β)+op(n-1/2).
By (35), we can prove Γˇ→𝒫Γ by the law of large numbers. Together with Lemma 8 and Slutsky’s theorem, this proves Theorem 2.



---

*Source: 807135-2013-12-29.xml*