# Empirical Likelihood for Multidimensional Linear

Authors: Liping Zhu
Journal: Journal of Probability and Statistics
 (2012)

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## Abstract

Imputation is a popular technique for handling missing data especially for plenty
of missing values. Usually, the empirical log-likelihood ratio statistic under imputation
is asymptotically scaled chi-squared because the imputing data are not i.i.d.
Recently, a bias-corrected technique is used to study linear regression model with
missing response data, and the resulting empirical likelihood ratio is asymptotically
chi-squared. However, it may suffer from the “the curse of high dimension” in multidimensional
linear regression models for the nonparametric estimator of selection
probability function. In this paper, a parametric selection probability function is
introduced to avoid the dimension problem. With the similar bias-corrected method,
the proposed empirical likelihood statistic is asymptotically chi-squared when the selection
probability is specified correctly and even asymptotically scaled chi-squared
when specified incorrectly. In addition, our empirical likelihood estimator is always
consistent whether the selection probability is specified correctly or not, and will
achieve full efficiency when specified correctly. A simulation study indicates that
the proposed method is comparable in terms of coverage probabilities.


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## Body

## 1. Introduction

Consider the following multidimensional linear regression model:(1.1)Yi=Xi′β+ϵi,1≤i≤n,
where Yi is a scalar response variable, Xi is a p×1 vector of design variable, β is a p×1 vector of regression parameter, and the errors ϵi are independent random variables with E[ϵi∣Xi]=0, Var[ϵi∣Xi]=σ2. Suppose that we have incomplete observations (Xi,Yi,δi), i=1,2,…,n, from this model, where all the Xi′s are observed, and δi=0 if Yi is missing, δi=1 otherwise. Throughout this paper, we assume that Y is missing at random (MAR); that is, the probability that Y is missing may depend on X but not on, Y that is, P(δ=1∣X,Y)=P(δ=1∣X). We focus on constructing confidence regions on β with the incomplete data (Xi,Yi,δi), i=1,2,…,n.The empirical likelihood (EL) method, introduced by Owen [1], has many advantages over normal approximation methods for constructing confidence intervals [1]. One is that it produces confidence intervals and regions whose shape and orientation are determined entirely by the data; the other is that empirical likelihood regions are range preserving and transformation respecting. But it cannot directly to be used in missing responses situation. A natural method is to impute the predictor of Y based on the completely observed pairs, which is provided by Wang and Rao [2]. Unfortunately, the empirical log-likelihood ratio under this imputation is always asymptotically scaled chi-squared. Therefore, the empirical log-likelihood ratio cannot be applied directly to make a statistical inference on the parameter.Recently, a bias-corrected technique combined imputation method, and Horvitz-Thompson inverse-selection weighted method is separately explored by Xue [3] and Qin et al. [4]. The provided empirical likelihood ratios obey Wilks theorem and the empirical likelihood estimator is consistent. However, the true selection probability function is ordinarily unknown, its nonparametric estimator may suffer from “the curse of high dimensionality” in multidimensional linear regression models.To avoid “the curse of high dimensionality”, it is customary to suppose a parametric selection probability function, but the risk of misspecified function maybe exist. In this paper, we consider the following two situations, one is that the selection probability is specified correctly, the other is that specified incorrectly. To the best of our knowledge, this issue is rarely to be discussed. With the similar bias-corrected method, the provided empirical likelihood statistic is asymptotically chi-squared in the first situation and asymptotically weighted sum of chi-squared in the second situation. In addition, the following desired feature is worth mentioning. The auxiliary random vector to construct the empirical likelihood statistic is just the same form as that of proposed by Robins et al. [5]. The Robins’ estimator has the characteristic of “doubly robust”, that is, the estimator is asymptotically unbiased either the underlying missing data mechanism or the underlying regression function is correctly specified. Because the underlying regression function is asymptotically correct in our situation, our estimator is always consistent in both cases. Even our estimator can achieve asymptotically full efficiency in the first situation. From this point, it is feasible to use the parametric selection probability function to construct an EL statistic.The rest of the paper is organized as follows. In Section2, separately, the EL statistic for β is constructed and the asymptotically results are shown in the two situations. Section 3 reports some simulation results on the performance of the proposed EL confidence region. The proof of the main result is given in the Appendix.


## 2. Main Result

For simplicity, denote the true selection probability function byp0(x) and a specified probability distribution function by p(x,θ) for given θ, a p×1 unknown vector parameter. Thus, the first situation means p0(x)=p(x,θ0) for some θ0, where θ0 is the true parameter and the second situation means p0(x)≠p(x,θ) for any θ. And let β̂r be the least square estimator of β based on the completely observed pairs (Xi,Yi,δi=1), i=1,2,…,n that is,β̂r=(∑i=1nδiXiXi′)-1∑i=1nδiXiYi.


### 2.1. Empirical Likelihood

In this subsection, an empirical likelihood statistic is conducted, and then some asymptotic results are shown when the selection probability is specified correctly, that is,p0(x)=p(x,θ0) for some θ0.Since the design variableXi is observable for each subject, the maximum likelihood estimator θ̂ can be obtained by maximizing the likelihood function(2.1)L(θ)=∏i=1np(Xi,θ)δi(1-p(Xi,θ))1-δi.The following regularity assumptions onL(θ) are sufficiently strong to ensure both consistency and asymptotic normality of the maximum likelihood estimator. Suppose U(θ0) is some field of θ0 where θ0 is the true parameter:(C1)
∂lnLB(θ)/∂θ, ∂2lnLB(θ)/∂θ2 and ∂3lnLB(θ)/∂θ3 exist in U(θ0) for all X;(C2)
|∂3lnLB(θ)/∂θ3|≤H(X) in U(θ0), and EH(X)<∞;(C3)
Eθ0[lnLB′(θ)/lnLB(θ)]=0, Eθ0[lnLB′′(θ)/lnLB(θ)]=0, I(θ0)=Eθ0[lnLB′(θ)/lnLB(θ)]2>0.Then, we useY̌i=(δi/p(Xi,θ̂))Yi+(1-δi/p(Xi,θ̂))Xi′β̂r, i=1,2,…,n, as “complete” data set for Y to construct the auxiliary random vectors
(2.2)Zin*(β)=Xi(Y̌i-Xi′β)=Xi{δip(Xi,θ̂)(Yi-Xi′β)+(1-δip(Xi,θ̂))Xi′(β̂r-β)}.
Thus, an empirical log-likelihood ratio is defined as(2.3)ln*(β)=-2max⁡{∑i=1nlog⁡(nωi)∣ωi≥0,∑i=1nωi=1,∑i=1nωiZin*(β)=0}.
Further, the maximum empirical likelihood estimator β̃ of β is to maximize {-ln*(β)}.To ensure asymptotic results, the following assumptions are needed:(C4)
p(x,θ) is uniformly continued in U(θ0) for all X;(C5)
A,D1 is a positive definite matrix, where A=E{XXT} and D1=E{{1/p(X,θ0)}XX′ɛ2};(C6)
p(x,θ0) has bounded almost surely and infxp(x,θ0)>0;where condition (C4) is common for some selection probability function. Condition, (C5)-(C6) are necessary for asymptotic normality of the maximum empirical likelihood estimator.Theorem 2.1.
Suppose that conditions (C1)–(C6) above hold. Ifβ is the true parameter and p(x,θ) is specified correctly, then
(2.4)ln*(β)⟶Dχp2,
where χp2 means the chi-square variable with p degrees of freedom. →D represents the convergence in distribution.
Let theχp2(1-α) be the 1-α quartile of the χp2 for 0<α<1. Using Theorem 2.1, we obtain an approximate 1-α confidence region for β, defined by
(2.5){β̌∣ln*(β̌)≤χp2(1-α)}.
Theorem 2.1 can also be used to test the hypothesis H0:β=β0. One could reject H0 if ln*(β̌)>χp2(1-α).Remark 2.2.
In general, plug-in empirical likelihood will asymptotically lead to a sum of weightedχ12 variables with unknown weights for the EL statistics proposed. However, when p(x,θ) is specified correctly, the EL statistic with two plug-ins has the limiting distribution of χp2 which is due to the following reasons. Firstly, the bias-correction method, that is, the selection function as inverse weight will eliminate the influence by the β̂r. Secondly, the estimating function has special structure, that is, the influence of θ̂ will be also eliminated if θ is concluded in the denominator of function.Theorem 2.3.
Under Conditions (C1)–(C6), ifp(x,θ) is specified correctly, then
(2.6)n(β̂-β)⟶DN(0,Σ1),
where Σ1=A-1D1A-1.To apply Theorem2.3 to construct the confidence region of β, we give the estimator of Σ1, say Σ̂1=Â-1D̂1Â-1, where Â and D̂1 are defined by Â=(1/n)∑i=1nXiXiT, D̂1=(1/n)∑i=1n(1/p(X,θ̂))XiXiT(Yi-Xi′β̂r)2.It is easily proved thatΣ̂1 is a consistent estimator of Σ1. Thus, by Theorem 2.3, we have (2.7)Σ̂1-1/2n(β̂-β)⟶DN(0,Ip),
where Ip is an identity matrix of order p. Using (10.2d) in Arnold [6], we can obtain(2.8)(β̂-β)TnΣ̂1-1(β̂-β)⟶Dχp2.
Therefore, the confidence region of β can be constructed by using (2.8).


### 2.2. Adjusted Empirical Likelihood Whenp0(x)≠p(x,θ)

In this subsection, we also construct the empirical likelihood statistic and discuss some asymptotic results when the selection probability is specified incorrectly, that is,p0(x)≠p(x,θ) for any θ.Since the design variableXi is observable for each subject, the quasi-maximum likelihood estimator θ̃, other than the maximum likelihood estimator, can be obtained by maximizing the likelihood function(2.9)L(θ)=∏i=1np(Xi,θ)δi(1-p(Xi,θ))1-δi
under some regularity assumptions.The following regularity assumptions are sufficiently strong to ensure both consistency and asymptotic normality of the quasi-maximum likelihood estimator. Letu=(x,δ), Ω=(Rp,{0,1}), g(u)=p0(x)δ(1-p0(x))1-δ, f(u,θ)=p(x,θ)δ(1-p(x,θ))1-δ. It is natural that g(u) is the true density function of u, and f(u,θ) does not contain the true structure g(u). The Kullback-Leibler Information Criterion (KLIC) can be defined by I(g:f,θ)=E(log[g(U)/f(U,θ)]), here, and in what follows, expectations are taken with respect to the true distribution g(u). When expectations of the partial derivatives exist, we define the matrices A(θ)=E(∂2logf/∂θiθj), B(θ)=E((∂logf/∂θi)(∂logf/∂θj)).(C7)
g(u) is measurable on Ω.(C8)
f(u,θ) are measurable in u for every θ in Θ, a compact subset of a Rp, and continuous u for every θ in Θ.(C9)
(a)E[log(g(U))] exists and |logf(u,θ)|≤m(u) for all θ in Θ, where m is integrable with respect to g; (b) I(g:f,θ) has a unique minimum at θ* in Θ.(C10)
E((∂2logf)/(∂θi)),i=1,…,p, are measurable functions of u for each θ in Θ and continuously second-order differentiable functions of θ for each u in Ω.(C11)
|∂2logf/∂θiθj| and |(∂logf/∂θi)(∂logf/∂θj)|,i,j=1,…,p are dominated by functions integrable with respect to g for all u in Ω and θ in Θ.(C12)
(a)θ* is interior to Θ; (b) B(θ*) is nonsingular; (c) θ* is a regular point of A(θ*).Then, we useỸi=(δi/p(Xi,θ̃))Yi+(1-(δi/p(Xi,θ̃)))Xi′β̂r,i=1,2,…,n as “complete” data set for Y to construct the auxiliary random vectors(2.10)Zin(β)=Xi(Ỹi-Xi′β)=Xi{δip(Xi,θ̃)(Yi-Xi′β)+(1-δip(Xi,θ̃))Xi′(β̂r-β)}.
Thus, an empirical log-likelihood ratio is defined as(2.11)ln(β)=-2max⁡{∑i=1nlog⁡(nωi)∣ωi≥0,∑i=1nωi=1,∑i=1nωiZin(β)=0}.
And the maximum empirical likelihood estimator β̃ of β is to maximize {-ln(β)}.To ensure asymptotic results, the following assumptions are needed.(C4′)
p(x,θ) is uniformly continued in U(θ*) for all x;(C5′)
A,C,E,F,G,D1,D2 is a positive definite matrix, where A=E{XXT}, C=E[XXT(1-p0(X)/p(X,θ*))], E=E[p0(X)XXT], F=E[p0(X)XXTɛ2], G=E[(p0(X)/p(X,θ*))XXTɛ2], D2=E{(p0(X)/p2(X,θ*))XX′ɛ2}, and D3=D2+CE-1FE-1CT+2CE-1G;(C6′)
p0(x) has bounded almost surely and infxp0(x)>0,where condition (C4′) is common for some selection probability function. Condition (C5′)-(C6′) are necessary for asymptotic normality of the maximum empirical likelihood estimator.Theorem 2.4.
Suppose that conditions (C7)–(C12) and (C4′)–(C6′) above hold. Ifβ is the true parameter and p(x,θ) is specified incorrectly, then
(2.12)ln(β)⟶D∑i=1pω̃iχ1,i2,
where χ12 means the chi-square variable with 1 degrees of freedom. The weights ω̃i, i=1,2,…,p are the eigenvalues of matrix D3-1D2. →D represents the convergence in distribution.Letr(β)=p/tr(D3-1D2) be adjustment factor. Along the lines of [7], it is straightforward to show that r(β)Σi=1pω̃iχ1i2→Dχp2.LetSn1(β)=Σi=1nZin(β)ZinT(β),  Sn2(β)=(Σi=1nZin(β))(Σi=1nZin(β))T,  r̂(β)=tr(Ŝn2-1(β)Sn2(β))/tr(Ŝn1-1(β)Sn2(β)). We define an adjusted log-likelihood ratio by(2.13)ln,ad(β)=r̂(β)ln(β).Corollary 2.5.
Under the conditions of Theorem2.4, one has
(2.14)ln,ad(β)⟶Dχp2.
Let the χp2(1-α) be the 1-α quartile of the χp2 for 0<α<1. Using Corollary 2.5, we obtain an approximate 1-α confidence region for β, defined by
(2.15){β̌∣ln,ad(β̌)≤χp2(1-α)}.
Corollary 2.5 can also be used to test the hypothesis H0:β=β0. One could reject H0 if ln,ad(β̌)>χp2(1-α).Note thatr(β)→1, when p(x,θ) is close to correct one. Actually, the adjustment factor r(β) reflects information loss due to the misspecification of p(x,θ).Theorem 2.6.
Under the conditions of Theorem2.4, one has
(2.16)n(β̃-β)⟶DN(0,Σ2),
where Σ2=A-1D3A-1.To apply Theorem2.6 to construct the confidence region of β, we give the estimator of Σ2, say Σ̂2=Â-1D̂3Â-1, where Â and D̂3 are defined by Â=(1/n)∑i=1nXiXiT, Ĉ=(1/n)∑i=1nXiXiT(1-δi/p(Xi,θ*)), Ê=(1/n)∑i=1nδiXiXiT, F̂=(1/n)∑i=1nδiXiXiT(Yi-Xi′β̂r)2, Ĝ=(1/n)∑i=1n(δi/p(x,θ*))XiXiT(Yi-Xi′β̂r)2, D̂2=(1/n)∑i=1n(δi/p2(x,θ*))XiXiT(Yi-Xi′β̂r)2), D̂3=D̂2+ĈÊ-1F̂Ê-1ĈT+2ĈÊ-1Ĝ.It is easily proved thatΣ̂2 is a consistent estimator of Σ2. Thus, by Theorem 2.6, we have(2.17)Σ̂2-1/2n(β̃-β)⟶DN(0,Ip),
where Ip is an identity matrix of order p. Using (10.2d) in Arnold [6], we can obtain(2.18)(β̃-β)TnΣ̂2-1(β̃-β)⟶Dχp2.
Therefore, the confidence region of β can be constructed by using (2.18).Remark 2.7.
The estimator proposed by Robins in this situation is to solve(2.19)∑i=1nδip(Xi,θ̂)Xi(Yi-Xi′β)+(1-δip(Xi,θ̂))XiXi′(β̂r-β)=0.
Noted that E(X(Y-X′β)∣X)=XX′(β-β)=XX′[(β̂r-β)+(β-β̂r)]=XX′(β̂r-β)+op(1/n) that is, in that case the underlying regression function can asymptotically correctly specified. So whether missing data mechanism is specified correctly or not, the estimator is always consistent and the estimator can achieve asymptotic full efficiency when specified correctly.


## 2.1. Empirical Likelihood

In this subsection, an empirical likelihood statistic is conducted, and then some asymptotic results are shown when the selection probability is specified correctly, that is,p0(x)=p(x,θ0) for some θ0.Since the design variableXi is observable for each subject, the maximum likelihood estimator θ̂ can be obtained by maximizing the likelihood function(2.1)L(θ)=∏i=1np(Xi,θ)δi(1-p(Xi,θ))1-δi.The following regularity assumptions onL(θ) are sufficiently strong to ensure both consistency and asymptotic normality of the maximum likelihood estimator. Suppose U(θ0) is some field of θ0 where θ0 is the true parameter:(C1)
∂lnLB(θ)/∂θ, ∂2lnLB(θ)/∂θ2 and ∂3lnLB(θ)/∂θ3 exist in U(θ0) for all X;(C2)
|∂3lnLB(θ)/∂θ3|≤H(X) in U(θ0), and EH(X)<∞;(C3)
Eθ0[lnLB′(θ)/lnLB(θ)]=0, Eθ0[lnLB′′(θ)/lnLB(θ)]=0, I(θ0)=Eθ0[lnLB′(θ)/lnLB(θ)]2>0.Then, we useY̌i=(δi/p(Xi,θ̂))Yi+(1-δi/p(Xi,θ̂))Xi′β̂r, i=1,2,…,n, as “complete” data set for Y to construct the auxiliary random vectors
(2.2)Zin*(β)=Xi(Y̌i-Xi′β)=Xi{δip(Xi,θ̂)(Yi-Xi′β)+(1-δip(Xi,θ̂))Xi′(β̂r-β)}.
Thus, an empirical log-likelihood ratio is defined as(2.3)ln*(β)=-2max⁡{∑i=1nlog⁡(nωi)∣ωi≥0,∑i=1nωi=1,∑i=1nωiZin*(β)=0}.
Further, the maximum empirical likelihood estimator β̃ of β is to maximize {-ln*(β)}.To ensure asymptotic results, the following assumptions are needed:(C4)
p(x,θ) is uniformly continued in U(θ0) for all X;(C5)
A,D1 is a positive definite matrix, where A=E{XXT} and D1=E{{1/p(X,θ0)}XX′ɛ2};(C6)
p(x,θ0) has bounded almost surely and infxp(x,θ0)>0;where condition (C4) is common for some selection probability function. Condition, (C5)-(C6) are necessary for asymptotic normality of the maximum empirical likelihood estimator.Theorem 2.1.
Suppose that conditions (C1)–(C6) above hold. Ifβ is the true parameter and p(x,θ) is specified correctly, then
(2.4)ln*(β)⟶Dχp2,
where χp2 means the chi-square variable with p degrees of freedom. →D represents the convergence in distribution.
Let theχp2(1-α) be the 1-α quartile of the χp2 for 0<α<1. Using Theorem 2.1, we obtain an approximate 1-α confidence region for β, defined by
(2.5){β̌∣ln*(β̌)≤χp2(1-α)}.
Theorem 2.1 can also be used to test the hypothesis H0:β=β0. One could reject H0 if ln*(β̌)>χp2(1-α).Remark 2.2.
In general, plug-in empirical likelihood will asymptotically lead to a sum of weightedχ12 variables with unknown weights for the EL statistics proposed. However, when p(x,θ) is specified correctly, the EL statistic with two plug-ins has the limiting distribution of χp2 which is due to the following reasons. Firstly, the bias-correction method, that is, the selection function as inverse weight will eliminate the influence by the β̂r. Secondly, the estimating function has special structure, that is, the influence of θ̂ will be also eliminated if θ is concluded in the denominator of function.Theorem 2.3.
Under Conditions (C1)–(C6), ifp(x,θ) is specified correctly, then
(2.6)n(β̂-β)⟶DN(0,Σ1),
where Σ1=A-1D1A-1.To apply Theorem2.3 to construct the confidence region of β, we give the estimator of Σ1, say Σ̂1=Â-1D̂1Â-1, where Â and D̂1 are defined by Â=(1/n)∑i=1nXiXiT, D̂1=(1/n)∑i=1n(1/p(X,θ̂))XiXiT(Yi-Xi′β̂r)2.It is easily proved thatΣ̂1 is a consistent estimator of Σ1. Thus, by Theorem 2.3, we have (2.7)Σ̂1-1/2n(β̂-β)⟶DN(0,Ip),
where Ip is an identity matrix of order p. Using (10.2d) in Arnold [6], we can obtain(2.8)(β̂-β)TnΣ̂1-1(β̂-β)⟶Dχp2.
Therefore, the confidence region of β can be constructed by using (2.8).


## 2.2. Adjusted Empirical Likelihood Whenp0(x)≠p(x,θ)

In this subsection, we also construct the empirical likelihood statistic and discuss some asymptotic results when the selection probability is specified incorrectly, that is,p0(x)≠p(x,θ) for any θ.Since the design variableXi is observable for each subject, the quasi-maximum likelihood estimator θ̃, other than the maximum likelihood estimator, can be obtained by maximizing the likelihood function(2.9)L(θ)=∏i=1np(Xi,θ)δi(1-p(Xi,θ))1-δi
under some regularity assumptions.The following regularity assumptions are sufficiently strong to ensure both consistency and asymptotic normality of the quasi-maximum likelihood estimator. Letu=(x,δ), Ω=(Rp,{0,1}), g(u)=p0(x)δ(1-p0(x))1-δ, f(u,θ)=p(x,θ)δ(1-p(x,θ))1-δ. It is natural that g(u) is the true density function of u, and f(u,θ) does not contain the true structure g(u). The Kullback-Leibler Information Criterion (KLIC) can be defined by I(g:f,θ)=E(log[g(U)/f(U,θ)]), here, and in what follows, expectations are taken with respect to the true distribution g(u). When expectations of the partial derivatives exist, we define the matrices A(θ)=E(∂2logf/∂θiθj), B(θ)=E((∂logf/∂θi)(∂logf/∂θj)).(C7)
g(u) is measurable on Ω.(C8)
f(u,θ) are measurable in u for every θ in Θ, a compact subset of a Rp, and continuous u for every θ in Θ.(C9)
(a)E[log(g(U))] exists and |logf(u,θ)|≤m(u) for all θ in Θ, where m is integrable with respect to g; (b) I(g:f,θ) has a unique minimum at θ* in Θ.(C10)
E((∂2logf)/(∂θi)),i=1,…,p, are measurable functions of u for each θ in Θ and continuously second-order differentiable functions of θ for each u in Ω.(C11)
|∂2logf/∂θiθj| and |(∂logf/∂θi)(∂logf/∂θj)|,i,j=1,…,p are dominated by functions integrable with respect to g for all u in Ω and θ in Θ.(C12)
(a)θ* is interior to Θ; (b) B(θ*) is nonsingular; (c) θ* is a regular point of A(θ*).Then, we useỸi=(δi/p(Xi,θ̃))Yi+(1-(δi/p(Xi,θ̃)))Xi′β̂r,i=1,2,…,n as “complete” data set for Y to construct the auxiliary random vectors(2.10)Zin(β)=Xi(Ỹi-Xi′β)=Xi{δip(Xi,θ̃)(Yi-Xi′β)+(1-δip(Xi,θ̃))Xi′(β̂r-β)}.
Thus, an empirical log-likelihood ratio is defined as(2.11)ln(β)=-2max⁡{∑i=1nlog⁡(nωi)∣ωi≥0,∑i=1nωi=1,∑i=1nωiZin(β)=0}.
And the maximum empirical likelihood estimator β̃ of β is to maximize {-ln(β)}.To ensure asymptotic results, the following assumptions are needed.(C4′)
p(x,θ) is uniformly continued in U(θ*) for all x;(C5′)
A,C,E,F,G,D1,D2 is a positive definite matrix, where A=E{XXT}, C=E[XXT(1-p0(X)/p(X,θ*))], E=E[p0(X)XXT], F=E[p0(X)XXTɛ2], G=E[(p0(X)/p(X,θ*))XXTɛ2], D2=E{(p0(X)/p2(X,θ*))XX′ɛ2}, and D3=D2+CE-1FE-1CT+2CE-1G;(C6′)
p0(x) has bounded almost surely and infxp0(x)>0,where condition (C4′) is common for some selection probability function. Condition (C5′)-(C6′) are necessary for asymptotic normality of the maximum empirical likelihood estimator.Theorem 2.4.
Suppose that conditions (C7)–(C12) and (C4′)–(C6′) above hold. Ifβ is the true parameter and p(x,θ) is specified incorrectly, then
(2.12)ln(β)⟶D∑i=1pω̃iχ1,i2,
where χ12 means the chi-square variable with 1 degrees of freedom. The weights ω̃i, i=1,2,…,p are the eigenvalues of matrix D3-1D2. →D represents the convergence in distribution.Letr(β)=p/tr(D3-1D2) be adjustment factor. Along the lines of [7], it is straightforward to show that r(β)Σi=1pω̃iχ1i2→Dχp2.LetSn1(β)=Σi=1nZin(β)ZinT(β),  Sn2(β)=(Σi=1nZin(β))(Σi=1nZin(β))T,  r̂(β)=tr(Ŝn2-1(β)Sn2(β))/tr(Ŝn1-1(β)Sn2(β)). We define an adjusted log-likelihood ratio by(2.13)ln,ad(β)=r̂(β)ln(β).Corollary 2.5.
Under the conditions of Theorem2.4, one has
(2.14)ln,ad(β)⟶Dχp2.
Let the χp2(1-α) be the 1-α quartile of the χp2 for 0<α<1. Using Corollary 2.5, we obtain an approximate 1-α confidence region for β, defined by
(2.15){β̌∣ln,ad(β̌)≤χp2(1-α)}.
Corollary 2.5 can also be used to test the hypothesis H0:β=β0. One could reject H0 if ln,ad(β̌)>χp2(1-α).Note thatr(β)→1, when p(x,θ) is close to correct one. Actually, the adjustment factor r(β) reflects information loss due to the misspecification of p(x,θ).Theorem 2.6.
Under the conditions of Theorem2.4, one has
(2.16)n(β̃-β)⟶DN(0,Σ2),
where Σ2=A-1D3A-1.To apply Theorem2.6 to construct the confidence region of β, we give the estimator of Σ2, say Σ̂2=Â-1D̂3Â-1, where Â and D̂3 are defined by Â=(1/n)∑i=1nXiXiT, Ĉ=(1/n)∑i=1nXiXiT(1-δi/p(Xi,θ*)), Ê=(1/n)∑i=1nδiXiXiT, F̂=(1/n)∑i=1nδiXiXiT(Yi-Xi′β̂r)2, Ĝ=(1/n)∑i=1n(δi/p(x,θ*))XiXiT(Yi-Xi′β̂r)2, D̂2=(1/n)∑i=1n(δi/p2(x,θ*))XiXiT(Yi-Xi′β̂r)2), D̂3=D̂2+ĈÊ-1F̂Ê-1ĈT+2ĈÊ-1Ĝ.It is easily proved thatΣ̂2 is a consistent estimator of Σ2. Thus, by Theorem 2.6, we have(2.17)Σ̂2-1/2n(β̃-β)⟶DN(0,Ip),
where Ip is an identity matrix of order p. Using (10.2d) in Arnold [6], we can obtain(2.18)(β̃-β)TnΣ̂2-1(β̃-β)⟶Dχp2.
Therefore, the confidence region of β can be constructed by using (2.18).Remark 2.7.
The estimator proposed by Robins in this situation is to solve(2.19)∑i=1nδip(Xi,θ̂)Xi(Yi-Xi′β)+(1-δip(Xi,θ̂))XiXi′(β̂r-β)=0.
Noted that E(X(Y-X′β)∣X)=XX′(β-β)=XX′[(β̂r-β)+(β-β̂r)]=XX′(β̂r-β)+op(1/n) that is, in that case the underlying regression function can asymptotically correctly specified. So whether missing data mechanism is specified correctly or not, the estimator is always consistent and the estimator can achieve asymptotic full efficiency when specified correctly.


## 3. Simulation

Due to the curse of nonparametric estimation, Xue’s method may be hard to realize. Here we conducted an extensive simulation study to compare the performances of the weighted-corrected empirical likelihood(WCEL) proposed in this paper and Wang’s method (AEL) proposed in Wang and Rao under the covariates of four dimensions.We considered the linear model (1.1) with d=4 and β=(0.8,1.5,1,2), where X was generated from a four-dimensions standard normal distribution, and ɛ was generated from the normal distribution with mean zero and variance 0.04.In the first case, the real selection probability functionp0(x,θ) was taken to be exp(x′θ)/(1+exp(x′θ)). We considered θ equaled to the three following values θ1=(-0.5,-0.5,-0.5,-0.5), θ2=(0,0,0,0), θ3=(0.5,0.5,0.5,0.5), respectively.In the second case, the real selection probability functionp0(x) was taken to be the following three cases:Case 1.
p1(x)=0.8+0.2(|x1|+|x2|+|x3|+|x4|) if |x1|+|x2|+|x3|+|x4|≤1, and 0.9 elsewhere.Case 2.
p2(x)=0.9-0.1(|x1|+|x2|+|x3|+|x4|)| if |x1|+|x2|+|x3|+|x4|≤4, and 0.9 elsewhere.Case 3.
p3(x)=0.6 for all x.We generated 5000 Monte Carlo random samples of sizen=100, 200, and 500 based on the above six selection probability functions p(x). When the working model was p(x,θ)=exp(x′θ)/(1+exp(x′θ)), the empirical coverage probabilities for β, with a nominal level 0.95, were computed based on the above two methods with 5000 simulation runs. The results are reported in Table 1.Table 1
Empirical coverage probabilities of the confidence regions forβ under different selection probability functions p(x) and sample sizes n when nominal level is 0.95.
nCase ICase IIp0(x,θ1)p0(x,θ2)p0(x,θ3)p1(x)p2(x)p3(x)1000.5540.610.5710.8610.7720.693WCEL2000.5940.7290.6110.9710.9160.8185000.6990.9140.6980.9950.9870.9671000.4860.4890.5160.6670.5910.521AEL2000.5490.5360.5640.8050.5870.5875000.6550.6530.6600.9320.8630.759From Table1, we can obtain the following results. Firstly, under both cases, WCEL performs better than AEL because its confidence regions have uniformly higher coverage probabilities. Secondly, all the empirical coverage probabilities increase as n increases for every fixed missing rate. Observably, the missing rate also affects coverage probability. Generally, the coverage probability decreases as the missing rate increases for every fixed sample size. However, under Case I, the values do hardly change by a large amount for both methods because of the exponential selection probability function.


## 4. Conclusion

In this paper, a parametric selection probability function is introduced to avoid the dimension difficulty, and a bias-corrected technique leads to an empirical likelihood (EL) statistic with asymptotically chi-square distribution when the selection probability specified correctly and with asymptotically weighted chi-square distribution when specified incorrectly. Also, our estimator is always consistent and will achieve asymptotic full efficiency when selection probability function is specified correctly.



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*Source: 473932-2012-03-27.xml*