Empirical Likelihood for Multidimensional Linear Model with Missing Responses

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.

1. Introduction

Consider the following multidimensional linear regression model:𝑌_𝑖=𝑋^𝑖𝛽+𝜖_𝑖,1≤𝑖≤𝑛,(1.1) where 𝑌_𝑖 is a scalar response variable, 𝑋_𝑖 is a 𝑝×1 vector of design variable, 𝛽 is a 𝑝×1 vector of regression parameter, and the errors 𝜖_𝑖 are independent random variables with 𝐸[𝜖_𝑖∣𝑋_𝑖]=0, Var[𝜖_𝑖∣𝑋_𝑖]=𝜎². Suppose that we have incomplete observations (𝑋_𝑖,𝑌_𝑖,𝛿_𝑖), 𝑖=1,2,…,𝑛, from this model, where all the 𝑋^𝑖𝑠 are observed, and 𝛿_𝑖=0 if 𝑌_𝑖 is missing, 𝛿_𝑖=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 (𝑋_𝑖,𝑌_𝑖,𝛿_𝑖), 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 Section 2, 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 by 𝑝₀(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 𝛽̂_𝑟 be the least square estimator of 𝛽 based on the completely observed pairs (𝑋_𝑖,𝑌_𝑖,𝛿_𝑖=1), 𝑖=1,2,…,𝑛 that is,𝛽̂_𝑟=(∑^𝑛𝑖=1𝛿_𝑖𝑋_𝑖𝑋^𝑖)⁻¹∑^𝑛𝑖=1𝛿_𝑖𝑋_𝑖𝑌_𝑖.

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 variable 𝑋_𝑖 is observable for each subject, the maximum likelihood estimator 𝜃̂ can be obtained by maximizing the likelihood function𝐿(𝜃)=^𝑛_𝑖=1𝑝𝑋_𝑖,𝜃^𝛿𝑖1−𝑝𝑋_𝑖,𝜃^1−𝛿𝑖.(2.1)

The following regularity assumptions on 𝐿(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)𝜕ln𝐿_𝐵(𝜃)/𝜕𝜃, 𝜕²ln𝐿_𝐵(𝜃)/𝜕𝜃² and 𝜕³ln𝐿_𝐵(𝜃)/𝜕𝜃³ exist in 𝑈(𝜃₀) for all 𝑋;(C2)|𝜕³ln𝐿_𝐵(𝜃)/𝜕𝜃³|≤𝐻(𝑋) in 𝑈(𝜃₀), and EH(𝑋)<∞;(C3)𝐸_𝜃0[ln𝐿^′_𝐵(𝜃)/ln𝐿_𝐵(𝜃)]=0, 𝐸_𝜃0[ln𝐿^𝐵(𝜃)/ln𝐿_𝐵(𝜃)]=0, 𝐼(𝜃₀)=𝐸_𝜃0[ln𝐿^𝐵(𝜃)/ln𝐿_𝐵(𝜃)]²>0.

Then, we use 𝑌_𝑖=(𝛿_𝑖/𝑝(𝑋_𝑖,𝜃̂))𝑌_𝑖+(1−𝛿_𝑖/𝑝(𝑋_𝑖,𝜃̂))𝑋^𝑖𝛽̂_𝑟, 𝑖=1,2,…,𝑛, as “complete” data set for 𝑌 to construct the auxiliary random vectors 𝑍^∗_in(𝛽)=𝑋_𝑖𝑌^𝑖−𝑋^′_𝑖𝛽=𝑋_𝑖𝑝𝑋𝛿^𝑖_𝑖,𝜃̂𝑌^𝑖−𝑋^𝑖𝛽+1−𝛿^𝑖𝑝𝑋_𝑖,𝜃̂𝑋^𝑖𝛽̂_𝑟−𝛽.(2.2) Thus, an empirical log-likelihood ratio is defined as𝑙^∗𝑛(𝛽)=−2max^𝑛_𝑖=1log𝑛𝜔_𝑖∣𝜔_𝑖≥0,^𝑛_𝑖=1𝜔_𝑖=1,^𝑛_𝑖=1𝜔_𝑖𝑍^∗_in(𝛽)=0.(2.3) Further, the maximum empirical likelihood estimator 𝛽̃ of 𝛽 is to maximize {−𝑙^∗𝑛(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{𝑋𝑋^𝑇} and 𝐷₁=𝐸{{1/𝑝(𝑋,𝜃₀)}𝑋𝑋^′𝜀²};(C6)𝑝(𝑥,𝜃₀) has bounded almost surely and in𝑓_𝑥𝑝(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. If 𝛽 is the true parameter and 𝑝(𝑥,𝜃) is specified correctly, then 𝑙^∗𝑛(𝛽)⟶^𝐷𝜒^2𝑝,(2.4) where 𝜒^2𝑝 means the chi-square variable with 𝑝 degrees of freedom. →^𝐷 represents the convergence in distribution.
Let the 𝜒^2𝑝(1−𝛼) be the 1−𝛼 quartile of the 𝜒^2𝑝 for 0<𝛼<1. Using Theorem 2.1, we obtain an approximate 1−𝛼 confidence region for 𝛽, defined by 𝛽̌∣𝑙^∗𝑛𝛽̌≤𝜒^2𝑝(1−𝛼).(2.5) Theorem 2.1 can also be used to test the hypothesis 𝐻₀∶𝛽=𝛽₀. One could reject 𝐻₀ if 𝑙^∗𝑛(𝛽̌)>𝜒^2𝑝(1−𝛼).

Remark 2.2. In general, plug-in empirical likelihood will asymptotically lead to a sum of weighted 𝜒²¹ 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 𝜒^2𝑝 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 𝛽̂_𝑟. 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), if 𝑝(𝑥,𝜃) is specified correctly, then √𝑛𝛽̂−𝛽⟶^𝐷𝑁0,Σ₁,(2.6) where Σ₁=𝐴⁻¹𝐷₁𝐴⁻¹.

To apply Theorem 2.3 to construct the confidence region of 𝛽, we give the estimator of Σ₁, say Σ₁=𝐴⁻¹𝐷₁𝐴⁻¹, where 𝐴 and 𝐷₁ are defined by 𝐴=(1/𝑛)∑^𝑛𝑖=1𝑋_𝑖𝑋^𝑇𝑖, 𝐷₁=(1/𝑛)∑^𝑛𝑖=1(1/𝑝(𝑋,𝜃̂))𝑋_𝑖𝑋^𝑇𝑖(𝑌_𝑖−𝑋^𝑖𝛽̂_𝑟)².

It is easily proved that Σ₁ is a consistent estimator of Σ₁. Thus, by Theorem 2.3, we have Σ^1−1/2√𝑛𝛽̂−𝛽⟶^𝐷𝑁0,𝐼_𝑝,(2.7) where 𝐼_𝑝 is an identity matrix of order 𝑝. Using (10.2d) in Arnold [6], we can obtain𝛽̂−𝛽^𝑇𝑛Σ¹⁻¹𝛽̂−𝛽⟶^𝐷𝜒^2𝑝.(2.8) Therefore, the confidence region of 𝛽 can be constructed by using (2.8).

2.2. Adjusted Empirical Likelihood When 𝑝₀(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 variable 𝑋_𝑖 is observable for each subject, the quasi-maximum likelihood estimator 𝜃̃, other than the maximum likelihood estimator, can be obtained by maximizing the likelihood function𝐿(𝜃)=^𝑛_𝑖=1𝑝𝑋_𝑖,𝜃^𝛿𝑖1−𝑝𝑋_𝑖,𝜃^1−𝛿𝑖(2.9) under some regularity assumptions.

The following regularity assumptions are sufficiently strong to ensure both consistency and asymptotic normality of the quasi-maximum likelihood estimator. Let 𝑢=(𝑥,𝛿), Ω=(𝑅^𝑝,{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/𝜕𝜃_i𝜃_𝑗), 𝐵(𝜃)=𝐸((𝜕log𝑓/𝜕𝜃_𝑖)(𝜕log𝑓/𝜕𝜃_𝑗)).(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 𝑅^𝑝, 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)/(𝜕𝜃_𝑖)),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/𝜕𝜃_𝑖𝜃_𝑗| and |(𝜕log𝑓/𝜕𝜃_𝑖)(𝜕log𝑓/𝜕𝜃_𝑗)|,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 use 𝑌_𝑖=(𝛿_𝑖/𝑝(𝑋_𝑖,𝜃̃))𝑌_𝑖+(1−(𝛿_𝑖/𝑝(𝑋_𝑖,𝜃̃)))𝑋^′_𝑖𝛽̂_𝑟,𝑖=1,2,…,𝑛 as “complete” data set for 𝑌 to construct the auxiliary random vectors𝑍_in(𝛽)=𝑋_𝑖𝑌^𝑖−𝑋^′_𝑖𝛽=𝑋_𝑖𝑝𝑋𝛿^𝑖_𝑖,𝜃̃𝑌ⁱ−𝑋^′_𝑖𝛽+1−𝛿^𝑖𝑝𝑋_𝑖,𝜃̃𝑋^𝑖𝛽̂_𝑟−𝛽.(2.10) Thus, an empirical log-likelihood ratio is defined as𝑙_𝑛(𝛽)=−2max^𝑛_𝑖=1log𝑛𝜔_𝑖∣𝜔_𝑖≥0,^𝑛_𝑖=1𝜔_𝑖=1,^𝑛_𝑖=1𝜔_𝑖𝑍_in(𝛽)=0.(2.11) And the maximum empirical likelihood estimator 𝛽̃ of 𝛽 is to maximize {−𝑙_𝑛(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{𝑋𝑋^𝑇}, 𝐶=𝐸[𝑋𝑋^𝑇(1−𝑝₀(𝑋)/𝑝(𝑋,𝜃^∗))], 𝐸=𝐸[𝑝₀(𝑋)𝑋𝑋^𝑇], 𝐹=𝐸[𝑝₀(𝑋)𝑋𝑋^𝑇𝜀²], 𝐺=𝐸[(𝑝₀(𝑋)/𝑝(𝑋,𝜃^∗))𝑋𝑋^𝑇𝜀²], 𝐷₂=𝐸{(𝑝₀(𝑋)/𝑝²(𝑋,𝜃^∗))𝑋𝑋^′𝜀²}, and 𝐷₃=𝐷₂+𝐶𝐸⁻¹𝐹𝐸⁻¹𝐶^𝑇+2𝐶𝐸⁻¹𝐺;(C6′)𝑝₀(𝑥) has bounded almost surely and 𝑖𝑛𝑓_𝑥𝑝₀(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. If 𝛽 is the true parameter and 𝑝(𝑥,𝜃) is specified incorrectly, then 𝑙_𝑛(𝛽)⟶^𝐷𝑝_𝑖=1𝜔_𝑖𝜒^21,𝑖,(2.12) where 𝜒²¹ means the chi-square variable with 1 degrees of freedom. The weights 𝜔_𝑖, 𝑖=1,2,…,𝑝 are the eigenvalues of matrix 𝐷³⁻¹𝐷₂. →^𝐷 represents the convergence in distribution.

Let 𝑟(𝛽)=𝑝/𝑡𝑟(𝐷³⁻¹𝐷₂) be adjustment factor. Along the lines of [7], it is straightforward to show that 𝑟(𝛽)Σ^𝑝𝑖=1𝜔_𝑖𝜒^2𝐷1𝑖→𝜒^2𝑝.

Let 𝑆_𝑛1(𝛽)=Σ^𝑛𝑖=1𝑍_in(𝛽)Z^𝑇_in(𝛽),  𝑆_𝑛2(𝛽)=(Σ^𝑛𝑖=1𝑍_in(𝛽))(Σ^𝑛𝑖=1𝑍_in(𝛽))^𝑇,  𝑟(̂𝛽)=𝑡𝑟(𝑆^−1𝑛2(𝛽)𝑆_𝑛2(𝛽))/𝑡𝑟(𝑆^−1𝑛1(𝛽)𝑆_𝑛2(𝛽)). We define an adjusted log-likelihood ratio by𝑙_𝑛,ad(𝛽)=𝑟̂(𝛽)𝑙_𝑛(𝛽).(2.13)

Corollary 2.5. Under the conditions of Theorem 2.4, one has 𝑙_𝑛,ad(𝛽)⟶^𝐷𝜒^2𝑝.(2.14) Let the 𝜒^2𝑝(1−𝛼) be the 1−𝛼 quartile of the 𝜒^2𝑝 for 0<𝛼<1. Using Corollary 2.5, we obtain an approximate 1−𝛼 confidence region for 𝛽, defined by 𝛽̌∣𝑙_𝑛,ad𝛽̌≤𝜒^2𝑝(1−𝛼).(2.15) Corollary 2.5 can also be used to test the hypothesis 𝐻₀∶𝛽=𝛽₀. One could reject 𝐻₀ if 𝑙_𝑛,ad(𝛽̌)>𝜒^2𝑝(1−𝛼).

Note that 𝑟(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 Theorem 2.4, one has √𝑛𝛽̃−𝛽⟶^𝐷𝑁0,Σ₂,(2.16) where Σ₂=𝐴⁻¹𝐷₃𝐴⁻¹.

To apply Theorem 2.6 to construct the confidence region of 𝛽, we give the estimator of Σ₂, say Σ₂=𝐴⁻¹𝐷₃𝐴⁻¹, where 𝐴 and 𝐷₃ are defined by 𝐴=(1/𝑛)∑^𝑛𝑖=1𝑋_𝑖𝑋^𝑇𝑖, 𝐶=(1/𝑛)∑^𝑛𝑖=1𝑋_𝑖𝑋^𝑇𝑖(1−𝛿_𝑖/𝑝(𝑋_𝑖,𝜃^∗)), 𝐸=(1/𝑛)∑^𝑛𝑖=1𝛿_𝑖𝑋_𝑖𝑋^𝑇𝑖, 𝐹=(1/𝑛)∑^𝑛𝑖=1𝛿_𝑖𝑋_𝑖𝑋^𝑇𝑖(𝑌_𝑖−𝑋^′_𝑖𝛽̂_𝑟)², 𝐺=(1/𝑛)∑^𝑛𝑖=1(𝛿_𝑖/𝑝(𝑥,𝜃^∗))𝑋_𝑖𝑋^𝑇𝑖(𝑌_𝑖−𝑋^𝑖𝛽̂_𝑟)², 𝐷₂=(1/𝑛)∑^𝑛𝑖=1(𝛿_𝑖/𝑝²(𝑥,𝜃^∗))𝑋_𝑖𝑋^𝑇𝑖(𝑌_𝑖−𝑋^𝑖𝛽̂_𝑟)²), 𝐷₃=𝐷₂+𝐶𝐸⁻¹𝐹𝐸⁻¹𝐶^𝑇+2𝐶𝐸⁻¹𝐺.

It is easily proved that Σ₂ is a consistent estimator of Σ₂. Thus, by Theorem 2.6, we haveΣ^2−1/2√𝑛𝛽̃−𝛽⟶^𝐷𝑁0,𝐼_𝑝,(2.17) where 𝐼_𝑝 is an identity matrix of order 𝑝. Using (10.2d) in Arnold [6], we can obtain𝛽̃−𝛽^𝑇𝑛Σ²⁻¹𝛽̃−𝛽⟶^𝐷𝜒^2𝑝.(2.18) 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 ^𝑛_𝑖=1𝛿_𝑖𝑝𝑋_𝑖,𝜃̂𝑋^𝑖𝑌_𝑖−𝑋^𝑖𝛽+1−𝛿^𝑖𝑝𝑋_𝑖,𝜃̂𝑋^𝑖𝑋^𝑖𝛽̂_𝑟−𝛽=0.(2.19) Noted that 𝐸(𝑋(𝑌−𝑋^𝛽)∣𝑋)=𝑋𝑋^(𝛽−𝛽)=𝑋𝑋^[(𝛽̂_𝑟−𝛽)+(𝛽−𝛽̂_𝑟)]=𝑋𝑋^(𝛽̂_𝑟−𝛽)+𝑜_𝑝(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 𝜀 was generated from the normal distribution with mean zero and variance 0.04.

In the first case, the real selection probability function 𝑝₀(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 function 𝑝₀(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 size 𝑛=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 for 𝛽 under different selection probability functions 𝑝(𝑥) and sample sizes 𝑛 when nominal level is 0.95.

From Table 1, 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.

Appendix

A. Proofs

Lemma A.1. Suppose that (C1)–(C4) hold. Then 1/𝑝(𝑋,𝜃̂)=(1/𝑝(𝑋,𝜃₀))(1+𝑜_𝑝(1)).

Proof of Lemma A.1. Under conditions (C1)–(C3), it can be shown that 𝜃̂ is √𝑛 consistent: 𝜃̂−𝜃₀=𝑂_𝑝𝑛^−1/2=𝑜_𝑝(1).(A.1) Together with condition (C4), we have 𝑝𝑋,𝜃̂=𝑝𝑋,𝜃₀+𝑜_𝑝(1).(A.2) By Taylor expanding, it is easily shown that 1𝑝𝑋,𝜃̂=𝑝𝑋1,𝜃₀1+𝑜^𝑝(1).(A.3)

Lemma A.2. Suppose that (C7)–(C12) hold. Then 1/𝑝(𝑋,𝜃̃)=(1/𝑝(𝑋,𝜃^∗))(1+𝑜_𝑝(1)).

Proof of Lemma A.2. From Theorem  3 of White [8], under conditions (C7)–(C12), it can be shown that 𝜃̃ is √𝑛 consistent: 𝜃̃−𝜃^∗=𝑂_𝑝𝑛^−1/2=𝑜_𝑝(1).(A.4) Similar to Lemma A.1, it is easy to show that 1/𝑝(𝑋,𝜃̃)=(1/𝑝(𝑋,𝜃^∗))(1+𝑜_𝑝(1)).

Lemma A.3. Suppose that (C7)–(C12) and (C4′)–(C6′) hold. If 𝑝(𝑋,𝜃) is specified wrongly, then 1√𝑛^𝑛_𝑖=1𝑍_in(𝛽)⟶𝑁0,𝐷₃,(A.5)1𝑛^𝑛_𝑖=1max𝑍||𝑍_in(_in𝛽)𝑍^𝑇_in(𝛽)⟶𝐷₂,(𝛽)||=𝑜𝑛^−1/2.(A.6)

Proof of Lemma A.3. We prove (A.5) only; (A.6) can be proved similarly.
By direct calculation, we have 1√𝑛^𝑛_𝑖=1𝑍_in(𝛽)=1√𝑛^𝑛_𝑖=1𝛿_𝑖𝑝𝑋_𝑖,𝜃̃𝑋^𝑖𝑌^𝑖−𝑋^′_𝑖𝛽+1√𝑛^𝑛_𝑖=11−𝑝𝑋𝛿^𝑖_𝑖,𝜃̃𝑋^𝑖=1𝑋^𝑖𝛽̂_𝑟−𝛽√𝑛^𝑛_𝑖=1𝛿_𝑖𝑝𝑋_𝑖,𝜃^∗𝑋^𝑖+1𝜀_𝑖1+𝑜_𝑝(1)𝑛^𝑛_𝑖=11−𝑝𝑋𝛿^𝑖_𝑖,𝜃^∗𝑋^𝑖𝑋^𝑖1𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝑋^𝑖⁻¹1√𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝜀_𝑖+1𝑛^𝑛_𝑖=1𝑋_𝑖𝛿_𝑖𝑝𝑋_𝑖,𝜃^∗𝑋^𝑖1𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝑋^𝑖⁻¹1√𝑛^𝑛_𝑖=1=1𝛿_𝑖𝑋_𝑖𝜀_𝑖𝑜_𝑝(1)√𝑛^𝑛_𝑖=1𝛿_𝑖𝑝X_𝑖,𝜃^∗𝑋^𝑖𝜀_𝑖+1𝑛^𝑛_𝑖=11−𝑝𝑋𝛿^𝑖_𝑖,𝜃^∗𝑋^𝑖𝑋^′_𝑖1𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝑋^𝑖⁻¹1√𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝜀_𝑖+𝑜_𝑝(1)(A.7) It is easily to shown that 1√𝑛^𝑛_𝑖=1𝛿_𝑖𝑝𝑋_𝑖,𝜃^∗𝑋^𝑖1𝜀_𝑖⟶𝑁0,𝐷₂,√𝑛^𝑛_𝑖=11𝛿_𝑖𝑋_𝑖𝜀_𝑖⟶𝑁(0,𝐹),𝑛^𝑛_𝑖=11−𝑝𝑋𝛿^𝑖_𝑖,𝜃^∗𝑋^𝑖1𝑋^𝑖=𝐶+𝑜_𝑝(1),𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝑋^𝑖=𝐸+𝑜_𝑝(1).(A.8) Note that 1𝑛^𝑛_𝑖=11−𝑝𝑋𝛿^𝑖_𝑖,𝜃^∗𝑋^𝑖𝑋^𝑖1𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝑋^𝑖⁻¹1√𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝜀_𝑖=1𝑛^𝑛_𝑖=11−𝑝𝑋𝛿^𝑖_𝑖,𝜃^∗𝑋^𝑖𝑋^𝑖−𝐶1𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝑋^𝑖⁻¹1√𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝜀_𝑖+𝐶⎛⎜⎜⎝1𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝑋^𝑖⁻¹−𝐸⁻¹⎞⎟⎟⎠1√𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝜀_𝑖+CE⁻¹1√𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝜀_𝑖=CE⁻¹1√𝑛^𝑛_𝑖=11𝛿_𝑖𝑋_𝑖𝜀_𝑖+𝑜_𝑝(1),√𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝜀_𝑖√1𝑛^𝑛_𝑖=1𝛿_𝑖𝑝𝑋_𝑖,𝜃^∗𝑋^𝑖𝜀_𝑖^′=𝐺+𝑜_𝑝(1),(A.9) so we have 1𝑛^𝑛_𝑖=11−𝑝𝑋𝛿^𝑖_𝑖,𝜃^∗𝑋^𝑖𝑋^𝑖1𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝑋^𝑖⁻¹1√𝑛^𝑛_𝑖=11𝛿_𝑖𝑋_𝑖𝜀_𝑖⟶𝑁0,𝐶𝐸⁻¹𝐹𝐸⁻¹𝐶^𝑇,𝑛^𝑛_𝑖=11−𝑝𝑋𝛿^𝑖_𝑖,𝜃^∗𝑋^𝑖𝑋^𝑖1𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝑋^𝑖⁻¹1√𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝜀_𝑖√1𝑛^𝑛_𝑖=1𝛿_𝑖𝑝𝑋_𝑖,𝜃^∗𝑋^𝑖𝜀_𝑖^=CE⁻¹1√𝑛^𝑛_𝑖=1𝛿_𝑖𝑋_𝑖𝜀_𝑖√1𝑛^𝑛_𝑖=1𝛿_𝑖𝑝𝑋_𝑖,𝜃^∗𝑋^𝑖𝜀_𝑖^′=CE₋₁+𝑜_𝑝(1)𝐺+𝑜_𝑝(1).(A.10) Therefore, by using (A.8)–(A.10), the proof of (A.5) is completed.

We just proof Theorems 2.4 and 2.6, the proofs of Theorems 2.1 and 2.3 are similar, only to replace the 𝜃̃,𝜃^∗,𝐷₃,𝑍_in(𝛽),𝐶 by 𝜃̂,𝜃₀,𝐷₁,𝑍^∗_in(𝛽),0 separately.

Proof of Theorem 2.4. By the Lagrange multiplier method, 𝑍_in(𝛽) can be represented as 𝑙_𝑛(𝜃)=2^𝑛_𝑖=1log1+𝜆^𝑇𝑍_in(𝛽),(A.11) where 𝜆=𝜆(𝛽) is a 𝑑×𝑎 vector given as the solution to ^𝑛_𝑖=1𝑍_in(𝛽)1+𝜆^𝑇𝑍_in(𝛽)=0,(A.12) by Lemma A.2, and using the same arguments as that of the proof (A.4) of in [9], we can show that ‖𝜆‖=𝑂_𝑝𝑛^−1/2.(A.13) Applying the Taylor expansion to (A.11) and (A.13), we get that 𝑙_𝑛(𝜃)=2^𝑛_𝑖=1𝜆^𝑇𝑍_in(𝛽)−𝜆^𝑇𝑍_in(𝛽)²2+𝑜(1).(A.14) By (A.12), it follows that 0=^𝑛_𝑖=1𝑍_in(𝛽)1+𝜆^𝑇𝑍_in(𝛽)=^𝑛_𝑖=1𝑍_in(𝛽)−^𝑛_𝑖=1𝑍_in(𝛽)𝜆^𝑇𝑍_in(𝛽)+𝑜_𝑝𝑛^−1/2.(A.15) This together with Lemma A.3 and (A.13) proves that ^𝑛_𝑖=1𝜆^𝑇𝑍_in(𝛽)²=^𝑛_𝑖=1𝜆=_𝑛𝜆^𝑇𝑍_in(𝛽)+𝑜(1),_𝑖=1𝑍_in(𝛽)𝑍^𝑇_in(𝛽)^𝑛−1_𝑖=1𝑍_in(𝛽)+𝑜_𝑝𝑛^−1/2.(A.16)
Therefore, from (A.14) we have 𝑙_𝑛(𝛽)=1√𝑛^𝑛_𝑖=1𝑍_in(𝛽)^𝑇1𝑛^𝑛_𝑖=1𝑍_in(𝛽)𝑍^𝑇_in(𝛽)⁻¹√1𝑛^𝑛_𝑖=1𝑍_in(𝛽)+𝑜(1).(A.17) This together with Lemma A.3 completes Theorem 2.4.

Proof of Theorem 2.6. From Theorem 1 of Qin [10] and (A.5), we obtain the result of Theorem 2.6 directly.

Acknowledgment

This paper is supported by NSF projects (2011YJYB005, 2011SSQD002 and 2011YJQT01) of Changji College Xinjiang Uygur Autonomous Region of China.