# Mean Empirical Likelihood Inference for Response Mean with Data Missing at Random

Authors: Hanji He; Guangming Deng
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

We extend the mean empirical likelihood inference for response mean with data missing at random. The empirical likelihood ratio confidence regions are poor when the response is missing at random, especially when the covariate is high-dimensional and the sample size is small. Hence, we develop three bias-corrected mean empirical likelihood approaches to obtain efficient inference for response mean. As to three bias-corrected estimating equations, we get a new set by producing a pairwise-mean dataset. The method can increase the size of the sample for estimation and reduce the impact of the dimensional curse. Consistency and asymptotic normality of the maximum mean empirical likelihood estimators are established. The finite sample performance of the proposed estimators is presented through simulation, and an application to the Boston Housing dataset is shown.


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## Body

## 1. Introduction

The missing data problems exist widely in the social sciences, political sciences, medical research, and many other fields. There are different missing patterns including single variable nonresponse, multivariate nonresponse, monotone nonresponse, and general nonresponse, and there are three types of missing mechanisms: missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR). The MAR is defined as missing that only depends on the observed. LetY be the value of a response that is subject to nonresponse and let X be a d-dimensional vector of auxiliary variables that are observed fully. Yo means the dataset of Y which is observed, and Ym means the dataset of Y which is missing or nonresponse. We interested in the mean of Y; i.e., μ=Ey. Let M be the indicator for Y, where M = 1 if Y is observed and M = 0 otherwise. In the MAR mechanism,(1)LA|Y,X=LA|Yo,X,means that missing data of Y is only related to Yo,X and has nothing to do with Ym.Empirical likelihood [1, 2] which is widely used for nonparametric and semiparametric statistical inferences is a competitive and powerful method for constructing confidence intervals (CIs). The empirical log-likelihood ratio (ELR) usually has an asymptotic chi-squared distribution, and the CI based on EL has many excellent properties, as proposed by Owen [3]. Due to these excellent properties, Wang and Rao [4], Liang et al. [5], and Stute et al. [6] extended the EL to missing data by imputing the missing data by a kernel regression function of observed data. For example, the bias-corrected EL method was constructed via the inverse propensity weighting (IPW), mean imputation (MI), and augmented inverse propensity weighting (AIPW). Guan et al. [7] proposed that an efficient estimator of the response means is constructed using the estimated response probabilities, which is a new method based on semiparametric maximum likelihood inference. Xie and Zhang [8] construct empirical likelihood confidence regions for the regression parameters without and with constraints by unconstrained and constrained empirical likelihood ratio statistics. Wang and Deng [9] proposed dimension-reduced empirical likelihood that was a two-stage estimation procedure and applied sufficient dimension reduction (SDR) technique in the kernel estimation of the propensity as well as the conditional mean response function. The method can avoid the well-known curse of dimensionality in the multivariate situation. But, it works only with a large sample. A lot of studies devote themselves to improve the accuracy of empirical likelihood ratio confidence region inference when there are small sample sizes and multidimensional situations. Hall and Scala [10], Diciccio et al. [11], and Tsao [12] found that empirical likelihood ratio confidence intervals could have poor accuracy, especially in the small sample and multidimensional situations. For confidence estimation, Dicicio et al. [11], Chen et al. [13], and Abraham and Wu [14] proposed the Bartlett correction empirical likelihood (BEL), the adjusted empirical likelihood, and an extended empirical likelihood (EEL), respectively. However, these methods require the calculation of the Bartlett correction constant, which is difficult to compute. So, Liang et al. [15] proposed the new method named mean empirical likelihood, which constructs an empirical likelihood function based on a set of pseudodata and it is easy to compute. Therefore, we need to develop a new empirical likelihood inference method for response mean with data missing at random in case of a small sample and multidimensional data.LetV be the pairwise-mean dataset. We will introduce a new empirical likelihood method named mean empirical likelihood. It constructs an empirical likelihood function based on a set of pseudodata and it is easy to complete compared to the mean empirical likelihood (MEL). The large sample properties of MEL are presented by Liang et al. [15]. The simulation studies indicate that the confidence regions by MEL are much more accurate than those by other empirical likelihood methods. Bigger coverage probability and smaller average interval length are other significant advantages of MEL. Nonetheless, there are two main problems in inference for response mean with data missing at random. First, the size of the sample is too small so that asymptotic normality is a little hard. The other is the high dimension of the covariate, and in these cases, the EL method is ineffective. MEL can solve the above problems by increasing the size of the sample for estimation.In this paper, we propose a new empirical likelihood to make efficient inference on response mean, especially for confidence region of response mean, by applying the mean empirical likelihood inference for small sample sizes and multidimensional variable situations. In specific, we first apply the mean empirical likelihood (MEL) method (see [15]) for response mean with data missing at random. We construct three bias-corrected nonparametric MEL functions by pairwise-mean dataset based on the IPW, MI, and AIPW approaches. We showed that the resulting three MEL methods yield asymptotically equivalent estimators that achieve the desirable asymptotic unbiasedness and asymptotic normality. After a factor adjusted to the MELR function based on the IPW or MI, the proposed three MELR functions are shown to be asymptotically standard chi-squared with (d − 1) degree of freedom, where d is the rank of the covariance matrix of the pair-mean dataset. Simulation results show that the proposed method not only has higher accurate coverage probabilities, smaller average lengths, and standard deviation but also has efficient point estimators when the sample size is only 40.This paper is organized as follows. Section2 presents three types of nonparametric MELRs. Then, we introduce our main idea and establish a number of asymptotic properties. The simulation studies are given in Section 3. Section 4 provides a real data analysis and the paper concludes with a discussion in Section 5.


## 2. Mean Empirical Likelihood Inference for Response Mean with Data Missing at Random

### 2.1. Some Methods for Missing Response Problem

LetXi,Yi,Mi,i=1,…,n, be independent and identically distributed realizations from X,Y,M, where Xi is a fully observed multivariate covariate, Yi is a univariate response having missing values, and Mi is a binary response indicator that equals 1 if and only if Yi is observed.In model (1), we make MAR assumptions about the full data, and the missing pattern is generally nonresponse. Furthermore, To adjust the weight, we introduce the probability function πX, which is πx=PrM=1|X,Y=PrM=1|X, and conditional mean response function, such that mX=EY|X,M=1=EMY|X/EM|X. The unspecified functions πX and mX can be estimated by the kernel regression estimators as follows:(2)π^X=∑j=1nMjKhX−Xj∑j=1nKhX−Xj,m^X=∑j=1nMjYjKhX−Xj∑j=1nδjKhX−Xj,where Kh•=h−1K•/h, K• is a symmetric kernel function, and h is a bandwidth.Next, we introduce three bias-corrected estimating equations using the nonparametric kernel estimatorsπ^X and m^X in equation (2) for handling missing data as follows.(i)
Nonparametric inverse propensity weighting (IPW) approach: this method assigns each observedYi with weight proportional to the inverse of the estimated propensity π^Xi in (2), i.e.,(3)φ^1iμ=MiYiπ^Xi−μ,i=1,…,n.(ii)
Nonparametric mean imputation (MI) approach: for eachXi with δi=0, we use an estimator m^Xi in equation (2) to estimate mXi, and a nonparametric imputation estimating equation for μ is given by(4)φ^2iμ=MiYi+1−Mim^Xi−μ,i=1,…,n.(iii)
Nonparametric augmented inverse propensity weighting (AIPW) approach: we combine the nonparametric IPW and MI approaches leading to the nonparametric AIPW estimating equation forμ as follows:(5)φ^3iμ=MiYiπ^Xi+1−Miπ^Xim^Xi−μ,i=1,…,n.In general, we use the empirical likelihood method to estimate the response mean. Letpli present the probability weight allocated to ϕ^liμ for l=1,2,3,i=1,…,n. The profile empirical log-likelihood ratio function for μ based on ϕ^liμ with data missing at random is defined as(6)R^lμ=−2sup∑i=1nlognpli|pli≥0,∑i=1npli=1,∑i=1npliϕ^liμ=0.However, when the sample size is less than 100 or the data is high dimensional, its finite sample properties may not work well because of the low precision ofχ2 approximation. Hence, the efficiency of the EL method for the response mean with data missing at random is low. In particular, the estimated coverage probability and average length are far from the theoretical value. The mean empirical likelihood can deal with a small sample and multidimensional situations of response mean problem with data missing at random.


### 2.2. Mean Empirical Likelihood for Response Mean with Data Missing at Random

In this paper, we apply the MEL to estimate the response mean with data missing at random under small sample and high-dimensional variate situation.We here follow notations in the previous section, and for simplicity, we denoteϕliμ=ϕliYi,μ=gXi,θ, i=1,…,n, and further denote the pairwise-mean dataset as follows:(7)ν=ϕliμ+ϕljμ2:1≤i≤j≤n,which can also be written as ν=Vl1μ,Vl2μ,…,VlNμ with N=nn+1/2. Let pli donate the probability weight allocated to ϕ^liμ for l=1,2,3,i=1,…,n. Based on this pairwise-mean dataset, the empirical log-likelihood ratio for μ is defined as(8)RlMμ=−2sup∑k=1NlogNplk|∑k=1NplkVlkμ=0,∑k=1Nplk=1,plk≥0,k=1,…,N,which is named mean empirical likelihood ratio. It follows that(9)RlMμ=−2∑k=1NlogNplk=2∑k=1Nlog1+λl′Vlkμ,where λ satisfying(10)1N∑k=1NVlkμ1+λl′Vlkμ=0,plk=1N1+λl′Vlkμ.Letθ0 be the unknown true parameter value. Then, the mean empirical log-likelihood ratio is given by(11)LlMμ0=−2logRlMμ0n+1.Now, we have the following main theorem.Theorem 1.
Under the conditions listed in the Appendix, asn⟶∞, we have(12)nμ^l−μ0⟶LN0,σ2,where l=1,2,3 denotes IPW, MI, and AIPW, respectively, and ⟶L denotes convergence in distribution.Theorem 2.
Under the conditions listed in the Appendix, asn⟶∞, we have(13)R^ipwμ0⟶L2ϑ1χ12,R^miμ0⟶L2ϑ2χ12,R^aipwμ0⟶L2χ12,where χ12 is the χ2-distributed random variable with one degree of freedom, ϑ1=σ2/σ12, ϑ2=σ2/σ22.Theorem 3.
Under the conditions thatCovϕiμ0=Σ exists and rankΣ=d, we have LMμ0⟶χ2d, in dist.Proof.
See Appendix.Following Theorem1, a confidence region for the parameter θ with asymptotic coverage probability 1−α can be defined as(14)IM=μ:LMμ≤χα2d.


## 2.1. Some Methods for Missing Response Problem

LetXi,Yi,Mi,i=1,…,n, be independent and identically distributed realizations from X,Y,M, where Xi is a fully observed multivariate covariate, Yi is a univariate response having missing values, and Mi is a binary response indicator that equals 1 if and only if Yi is observed.In model (1), we make MAR assumptions about the full data, and the missing pattern is generally nonresponse. Furthermore, To adjust the weight, we introduce the probability function πX, which is πx=PrM=1|X,Y=PrM=1|X, and conditional mean response function, such that mX=EY|X,M=1=EMY|X/EM|X. The unspecified functions πX and mX can be estimated by the kernel regression estimators as follows:(2)π^X=∑j=1nMjKhX−Xj∑j=1nKhX−Xj,m^X=∑j=1nMjYjKhX−Xj∑j=1nδjKhX−Xj,where Kh•=h−1K•/h, K• is a symmetric kernel function, and h is a bandwidth.Next, we introduce three bias-corrected estimating equations using the nonparametric kernel estimatorsπ^X and m^X in equation (2) for handling missing data as follows.(i)
Nonparametric inverse propensity weighting (IPW) approach: this method assigns each observedYi with weight proportional to the inverse of the estimated propensity π^Xi in (2), i.e.,(3)φ^1iμ=MiYiπ^Xi−μ,i=1,…,n.(ii)
Nonparametric mean imputation (MI) approach: for eachXi with δi=0, we use an estimator m^Xi in equation (2) to estimate mXi, and a nonparametric imputation estimating equation for μ is given by(4)φ^2iμ=MiYi+1−Mim^Xi−μ,i=1,…,n.(iii)
Nonparametric augmented inverse propensity weighting (AIPW) approach: we combine the nonparametric IPW and MI approaches leading to the nonparametric AIPW estimating equation forμ as follows:(5)φ^3iμ=MiYiπ^Xi+1−Miπ^Xim^Xi−μ,i=1,…,n.In general, we use the empirical likelihood method to estimate the response mean. Letpli present the probability weight allocated to ϕ^liμ for l=1,2,3,i=1,…,n. The profile empirical log-likelihood ratio function for μ based on ϕ^liμ with data missing at random is defined as(6)R^lμ=−2sup∑i=1nlognpli|pli≥0,∑i=1npli=1,∑i=1npliϕ^liμ=0.However, when the sample size is less than 100 or the data is high dimensional, its finite sample properties may not work well because of the low precision ofχ2 approximation. Hence, the efficiency of the EL method for the response mean with data missing at random is low. In particular, the estimated coverage probability and average length are far from the theoretical value. The mean empirical likelihood can deal with a small sample and multidimensional situations of response mean problem with data missing at random.


## 2.2. Mean Empirical Likelihood for Response Mean with Data Missing at Random

In this paper, we apply the MEL to estimate the response mean with data missing at random under small sample and high-dimensional variate situation.We here follow notations in the previous section, and for simplicity, we denoteϕliμ=ϕliYi,μ=gXi,θ, i=1,…,n, and further denote the pairwise-mean dataset as follows:(7)ν=ϕliμ+ϕljμ2:1≤i≤j≤n,which can also be written as ν=Vl1μ,Vl2μ,…,VlNμ with N=nn+1/2. Let pli donate the probability weight allocated to ϕ^liμ for l=1,2,3,i=1,…,n. Based on this pairwise-mean dataset, the empirical log-likelihood ratio for μ is defined as(8)RlMμ=−2sup∑k=1NlogNplk|∑k=1NplkVlkμ=0,∑k=1Nplk=1,plk≥0,k=1,…,N,which is named mean empirical likelihood ratio. It follows that(9)RlMμ=−2∑k=1NlogNplk=2∑k=1Nlog1+λl′Vlkμ,where λ satisfying(10)1N∑k=1NVlkμ1+λl′Vlkμ=0,plk=1N1+λl′Vlkμ.Letθ0 be the unknown true parameter value. Then, the mean empirical log-likelihood ratio is given by(11)LlMμ0=−2logRlMμ0n+1.Now, we have the following main theorem.Theorem 1.
Under the conditions listed in the Appendix, asn⟶∞, we have(12)nμ^l−μ0⟶LN0,σ2,where l=1,2,3 denotes IPW, MI, and AIPW, respectively, and ⟶L denotes convergence in distribution.Theorem 2.
Under the conditions listed in the Appendix, asn⟶∞, we have(13)R^ipwμ0⟶L2ϑ1χ12,R^miμ0⟶L2ϑ2χ12,R^aipwμ0⟶L2χ12,where χ12 is the χ2-distributed random variable with one degree of freedom, ϑ1=σ2/σ12, ϑ2=σ2/σ22.Theorem 3.
Under the conditions thatCovϕiμ0=Σ exists and rankΣ=d, we have LMμ0⟶χ2d, in dist.Proof.
See Appendix.Following Theorem1, a confidence region for the parameter θ with asymptotic coverage probability 1−α can be defined as(14)IM=μ:LMμ≤χα2d.


## 3. Simulation Studies

We present two simulation studies in this section, which correspond to the finite sample performance of different MEL methods inSection 2. We will compare the MEL with OEL for response mean with data missing at random. For the one-dimensional covariate model, the estimating equation gX,θ is equal to the estimating equation ϕliYi,μ. For the multivariate covariate model, we compare the performance for different πXi, correlation structures, and the size of the sample.


### 3.1. Simulation 1: Single Covariate Model

The results and discussion may be presented separately, or in one combined section, and may optionally be divided into headed subsections. Suppose thatgX,θ=ϕlY,μ=Y−μ is the estimating equation for μ. We aim to compare the confidence intervals derived from MEL and OEL, for a given sample size n. We consider different scenarios by generating observations X1,X2,…,Xn from N0,1, respectively.There are four indicators for evaluating method performance. The bias represents the difference between the estimated result and the true value, which measures the accuracy of point estimators. The coverage probability represents the probability that the point estimator falls within the confidence interval of the true value, which measures the stability of point estimators. The average length represents the average length of the confidence interval of the estimated value, and the variance represents the degree of dispersion of the estimated value, which measures the accuracy of interval estimators.Based on 5000 replicates, the bias, coverage proportions, average length, and standard deviation were calculated. The simulation results are presented in Table1.Table 1
Performance of the point estimators and 95% CIs for response mean in simulation 1,εi∼N0,1.
ModelOELMELμ^ipwμ^miμ^aipwY¯μ^ccμ^ipwμ^miμ^aipwY¯μ^ccn = 20Bias−0.01060.01520.00980.00000.0586−0.01060.01520.00980.00000.0586CP0.91150.91300.91550.91350.90850.92900.94700.93750.91350.9085AL1.27771.30511.29991.17711.42631.17641.26071.22121.17711.4263SD1.71541.18951.42980.54470.59781.24150.86091.03480.54470.5978n = 30Bias−0.00980.00880.0030−0.00570.0549−0.00980.00880.0030−0.00570.0549CP0.92850.92850.92950.92100.91250.94100.95300.94750.92100.9125AL1.07181.09311.08570.94701.14620.94330.99840.96800.94701.1462SD1.73301.21931.46890.48870.53671.24510.87601.05530.48870.5367n = 40Bias−0.00790.00760.0020−0.00560.0564−0.00790.00760.0020−0.00560.0564CP0.92950.92850.92650.92650.91250.93800.95300.94450.92650.9125AL0.93370.95010.94340.83891.01140.79890.83580.81400.83891.0114SD1.73191.22831.48130.46030.50491.23950.87911.06010.46030.5049Table1 shows the confidence region characteristics of different empirical likelihood methods when the Xi is the one-dimensional normal distribution with mean 1 and variate 1, denoted as Xi∼N1,1. Yi=Xi+εi with εi∼N0,1 for i=1,2,…,n. Then, we generate δi from the Bernoulli distribution with probability function πXi=1/1+exp−0.6−0.2Xi.From Tables1 and 2, we conclude the following:(1)
Comparison for different sample sizes: when the sample sizes increase, all coverage probability estimated by OEL or MEL increases for small sample sizen and all coverage probability estimated by OEL or MEL decreases. However, the bias estimated by AIPW and MI method reduced significantly, and the bias estimated by the complete-case and IPW method changed a little(2)
Comparison of different estimation methods: MEL is much better than OEL. All coverage probabilities estimated by MEL are closer to the nominal levels than those of the OEL method. All standard deviations by MEL are much smaller than those of the OEL method. All average lengths by MEL are much smaller than those of the OEL method(3)
Comparison for different estimating equations: the point estimator based on MI is superior to other estimating equations. In particular, the coverage probabilities estimated by the MI method perform very well in all simulations. In a word, the sequence about the accuracy of the estimating equation is that the MI method is superior to the AIPW method and the AIPW method is superior to IPW(4)
Comparison for different error situations: the average length estimated by OEL and MEL withεi∼N0,1+0.5X2 is much longer and more stable than the method with εi∼N0,1. But the standard deviation estimated by OEL and MEL with εi∼N0,1+0.5X2 is much bigger than the method with εi∼N0,1. Hence, when the errors in the method are heteroscedastic with εi∼N0,1+0.5X2, MEL method for response mean also plays wellTable 2
Performance of the point estimators and 95% CIs for response mean in simulation 1,εi∼N0,1+0.5X2.
ModelOELMELμ^ipwμ^miμ^aipwY¯μ^ccμ^ipwμ^miμ^aipwY¯μ^ccn = 20Bias−0.01030.01460.00850.00110.0588−0.01030.01460.00850.00110.0588CP0.90800.91550.91000.91750.92100.92800.93900.93650.91750.9210AL1.59121.63471.61771.43391.74871.47721.59651.52631.43391.7487SD2.02561.43451.76410.60030.66051.46621.03851.27700.60030.6605n = 30Bias−0.01010.00770.0025−0.00760.0508−0.01010.00770.0025−0.00760.0508CP0.92350.92750.92450.91450.91400.94000.95450.94400.91450.9140AL1.33171.36841.34921.15241.40581.17971.26511.20831.15241.4058SD2.05501.46871.80800.53850.59351.47641.05521.29900.53850.5935n = 40Bias−0.01370.0024−0.0041−0.00920.0503−0.01370.0024−0.0041−0.00920.0503CP0.92500.93000.92900.92800.91600.93550.95400.94400.92800.9160AL1.16361.19431.17651.01961.24211.00041.06371.01901.01961.2421SD2.05541.47991.82860.50700.55891.47101.05921.30870.50700.5589In a word, the performances of IPW, MI, and AIPW based on MEL precede those methods based on OEL, and AIPW based on MEL achieves the most accurate results.


### 3.2. Simulation 2: Multivariate Covariate Model

In simulation 2,Xi′s are 6-dimensional normal distribution with mean 0 and covariance matrix Γ, denoted by Xi∼N0,Γ, where Γ is a 6×6 symmetric matrix with Γjj=1 and Γjj′=ρj′−j for 1≤j<j′≤6, Yi=Xi1+Xi2+⋯+Xi6+εi with εi∼N0,σ2Xi for i=1,…,n. Two kinds of εi′s are considered:(i)
The first generates homoscedastic errors whereεi and Xi are independent and σ2Xi=1(ii)
The second errors are heteroscedastic withσ2Xi=1+0.5Xi12,i=1,2,…,nWe generateδi from the Bernoulli distribution with probability πXi and consider three choices of πXi:(15)M1.πXi=11+exp−0.6−0.2Xi1+Xi2+Xi3,M2.πXi=11+exp−0.2−0.4expXi1+Xi2+Xi3,M3.πXi=1−exp−exp0.1+0.2Xi1+Xi2+Xi3.The coefficients in the propensity models are chosen so that the unconditional rates of missing data are between 20% and 40%. WhenYi are linear and πXi is logistic linear under M1–M3, so SY and Sδ under M1–M3 are one-dimensional.In addition, we also conduct two simulations with four correlation structures:(i)
Independent covariates, where the correlation coefficient is 0(ii)
Slightly correlated covariates, where the correlation coefficient is 0.2(iii)
Moderately correlated covariates, where the correlation coefficient is 0.5(iv)
Strongly correlated covariates, where the correlation coefficient is 0.8All results are based on 1000 replications and the sample sizes aren = 20 and n = 40. Tables 3 and 4 show the biases and standard deviations (SDs) of the point estimators, the average lengths (ALs), and the coverage probabilities (CPs) of confidence intervals (CIs) at the nominal level of 95%. The CIs based on the mean empirical likelihood ratio LMμ are obtained by IMin equation (15). The CIs based on μ^=Y¯ and μcc are obtained by the normal approximation μ^−1.96σ^se,μ^+1.96σ^se, with standard error σse obtained by the square roots of bootstrap variance estimators based on n/4 bootstrap replications, where n is the size of the sample.Table 3
Performance of the point estimators and 95% CIs for response mean in simulation 2,εi∼N0,1.
ModelOELMELμ^ipwμ^miμ^aipwμ^ipwμ^miμ^aipwY¯μ^ccρ=0M1n = 20Bias0.08080.07350.05690.08080.07350.0569−0.00570.2179CP0.88300.88100.87300.90600.91600.90400.86000.8460AL2.23072.23622.22532.07562.14832.09392.08022.5970SD2.81282.15892.44512.03761.56671.77340.71450.7947n = 40Bias0.08020.05930.04650.08020.05930.04650.00050.2186CP0.92500.92200.92000.93600.93900.93400.87200.8380AL1.60841.60411.59731.37851.39561.37731.39591.7330SD2.87622.29722.51722.05851.64421.80170.58580.6519M2n = 20Bias0.12080.10800.08090.12080.10800.0809−0.00570.3595CP0.89600.88100.88600.92100.91200.91400.86000.7970AL2.26992.26692.26382.11302.15982.12652.08022.4904SD2.78702.27312.49812.01901.64881.81120.71450.7792n = 40Bias0.09040.06130.03340.09030.06130.03340.00050.3539CP0.93600.93500.92900.94600.94600.94200.87200.7700AL1.63111.62681.62211.39901.41021.39781.39591.6898SD2.85352.39502.56212.04231.71421.83380.58580.6434M3n = 20Bias0.11510.10580.08270.11510.10580.0827−0.00570.3179CP0.88800.88800.88100.90800.91100.90900.86000.8320AL2.23042.23182.22332.07552.13592.08972.08022.5149SD2.77512.18712.44852.01041.58671.77540.71450.7820n = 40Bias0.10630.07770.05740.10630.07770.05740.00050.3119CP0.91700.92000.92000.93400.93500.93500.87200.7810AL1.60661.60151.59591.37741.39131.37561.39591.6988SD2.84702.31502.51812.03761.65701.80230.58580.6447ρ=0.2M1n = 20Bias0.11540.10440.08410.11540.10440.08410.00540.3065CP0.87100.87900.87000.89900.90700.91100.86700.8330AL2.55722.54942.54342.38282.44502.39432.31662.9036SD3.21472.48592.80352.33031.80762.03660.75360.8401n = 40Bias0.08920.06460.04810.08920.06460.0481−0.00310.2807CP0.91400.90800.91200.93300.92500.92500.85300.8190AL1.83631.83341.82591.57281.59481.57421.58252.0088SD3.31332.62782.87882.37161.88112.06060.62380.7026M2n = 20Bias0.17130.15920.12470.17130.15920.12470.00540.4961CP0.89800.89900.90200.92000.92300.93000.86700.7940AL2.57562.57362.57062.39662.44662.41432.31662.7918SD3.16572.61162.83772.29431.89532.05840.75360.8257n = 40Bias0.09690.06860.03300.09690.06860.0330−0.00310.4320CP0.92700.92400.93000.93900.94500.94300.85300.7840AL1.86301.85991.85461.59741.61141.59781.58251.9375SD3.27852.75022.93052.34671.96862.09760.62380.6895M3n = 20Bias0.16170.14410.11800.16170.14410.11800.00540.4538CP0.87700.87300.87100.90500.90200.90500.86700.8020AL2.56282.55782.55362.38682.44662.40152.31662.8692SD3.19982.51642.81252.31961.82842.04160.75360.8361n = 40Bias0.13570.09980.07310.13570.09980.0731−0.00310.4151CP0.91300.91100.91400.92100.92900.93100.85300.7560AL1.83261.82681.82181.57091.58591.57051.58251.9471SD3.26522.65532.87392.33721.90072.05710.62380.6908ρ=0.5M1n = 20Bias0.15350.14110.11490.15350.14110.11490.00890.5182CP0.87100.87100.86900.90400.89600.90300.86000.8020AL3.21163.19633.19492.99263.05613.00662.95223.6533SD4.06803.15483.52972.95272.29922.56860.85110.9427n = 40Bias0.14130.10510.07870.14130.10510.0787−0.00120.4787CP0.90100.90700.90600.91600.92500.92100.85400.7730AL2.30342.29672.28961.97291.99551.97402.00392.5153SD4.20033.32003.61193.00742.37782.58650.70180.7849M2n = 20Bias0.20560.19010.14810.20560.19010.14810.00890.7143CP0.89500.89500.89900.92200.92000.92800.86000.7810AL3.26693.25753.25753.03653.08993.05462.95223.5512SD4.01943.34943.60822.91772.43592.62160.85110.9314n = 40Bias0.12600.09650.04690.12600.09650.0469−0.00120.6236CP0.93300.93000.93600.94300.94000.94200.85400.7430AL2.36122.35692.35152.02442.03982.02582.00392.4226SD4.17233.51643.71872.98722.51792.66250.70180.7707M3n = 20Bias0.31240.28020.22890.31240.28020.22890.01511.2031CP0.87300.85100.87500.90500.88300.90600.86900.7220AL4.14184.09184.11093.86003.87673.85643.86034.6516SD5.21214.15754.57683.79323.03983.33800.97351.0647n = 40Bias0.19270.13760.09250.19270.13760.0925−0.00120.6938CP0.91200.90700.91700.92200.92300.92900.85400.6960AL2.31482.30152.29771.98481.99651.98082.00392.4544SD4.17193.36483.62802.98722.40982.59790.70180.7748ρ=0.8M1n = 20Bias0.22080.21080.17870.22080.21080.17870.01510.8554CP0.86300.85100.86300.89600.89400.90300.86900.7790AL4.14784.11974.12333.86813.93673.87693.86034.7258SD5.26784.10584.58753.83293.00623.35010.97351.0731n = 40Bias0.20780.16580.12020.20780.16580.12020.00270.8162CP0.90000.90600.91300.92000.91900.92500.85600.7350AL2.98512.97302.96722.55762.58212.55932.60543.2279SD5.48544.31724.68553.93033.09543.35810.80000.8886M2n = 20Bias0.24860.23890.18060.24860.23890.18060.01511.0812CP0.89700.88100.89700.92400.90900.92200.86900.7530AL4.24644.21054.21973.95233.97853.96073.86034.5830SD5.27694.41564.71843.84043.22283.43850.97351.0572n = 40Bias0.16110.14310.07090.16110.14310.07090.00270.9679CP0.93000.92800.93300.94700.93600.95000.85600.6910AL3.07433.06443.06062.63222.65012.63642.60543.0973SD5.48674.61034.84603.93073.30393.47190.80000.8721M3n = 20Bias0.31240.28020.22890.31240.28020.22890.01511.2031CP0.87300.85100.87500.90500.88300.90600.86900.7220AL4.14184.09184.11093.86003.87673.85643.86034.6516SD5.21214.15754.57683.79323.03983.33800.97351.0647n = 40Bias0.29100.21930.14720.29100.21930.14720.00271.1758CP0.90400.90000.91100.92000.92200.93700.85600.6100AL3.00012.97232.97382.57392.57682.56442.60543.1358SD5.45744.36974.69883.91113.13283.36710.80000.8764Table 4
Performance of the point estimators and 95% CIs for response mean in simulation 2,εi∼N0,1+0.5Xi12.
ModelOELMELμ^ipwμ^miμ^aipwμ^ipwμ^miμ^aipwY¯μ^ccρ=0M1n = 20Bias0.09800.09550.08040.09800.09550.0804−0.00530.2177CP0.89000.88100.88400.91000.91200.91500.85300.8350AL2.35742.36342.35402.19892.27972.21962.15192.6703SD2.94162.23822.57612.13151.62581.86950.72640.8057n = 40Bias0.06060.04010.02680.06060.04010.02680.00190.2058CP0.92300.92100.91900.93300.93700.93300.86800.8320AL1.67241.66901.66131.43371.45391.43281.44101.8057SD2.97632.36702.61562.13021.69421.87210.59480.6646M2n = 20Bias0.12270.11170.08560.12270.11170.0856−0.00530.3586CP0.87900.87900.88100.90200.91500.90600.85300.8150AL2.35492.35682.34872.19402.24952.20702.15192.5743SD2.88802.34792.58982.09231.70411.87870.72640.7921n = 80Bias0.08110.05080.02430.08110.05080.02430.00190.3421CP0.93600.93100.93500.94400.94600.94700.86800.7940AL1.69701.69321.68761.45641.46971.45491.44101.7536SD2.95642.47052.66112.11591.76821.90460.59480.6549M3n = 20Bias0.13920.12660.10500.13920.12660.1050−0.00530.3263CP0.87800.87900.88100.91100.91500.91600.85300.8290AL2.34072.34312.33612.18202.24602.19912.15192.5981SD2.88942.26872.56502.09321.64661.86040.72640.7947n = 80Bias0.09180.06500.04370.09180.06500.04370.00190.3066CP0.92900.92800.93200.93900.94600.94300.86800.8030AL1.67501.66841.66291.43721.45161.43441.44101.7434SD2.95892.38672.61832.11771.70831.87400.59480.6531ρ=0.2M1n = 20Bias0.09910.09800.07890.09910.09800.0789−0.00570.2874CP0.87500.87300.87700.89700.91500.90400.85800.8330AL2.63302.63992.62972.45512.54502.47932.38122.9765SD3.30562.50422.87982.39651.82042.09150.76510.8508n = 40Bias0.06660.04520.02620.06660.04520.0262−0.00680.2749CP0.90200.91500.91100.91000.93500.92900.85800.8030AL1.90211.89721.89121.62991.65171.63161.66312.0389SD3.42672.69802.97572.45271.93132.13000.63890.7060M2n = 20Bias0.12650.11330.08430.12650.11330.0843−0.00570.4260CP0.89500.90000.90000.91900.92700.92500.85800.7880AL2.66132.65892.65222.47872.53092.49082.38122.8391SD3.25972.68072.93232.36361.94682.12830.76510.8321n = 40Bias0.08920.05670.02440.08920.05670.0244−0.00680.4145CP0.92300.92300.91900.93400.93500.93600.85800.7650AL1.93391.92891.92461.65911.67221.65901.66311.9615SD3.38372.83913.03652.42202.03232.17350.63890.6934M3n = 20Bias0.16460.14770.12370.16460.14770.1237−0.00570.4220CP0.88300.87300.87900.90200.91000.90700.85800.8010AL2.59712.59842.58882.42252.48592.43142.38122.8912SD3.20572.54332.84972.32431.84812.06900.76510.8394n = 40Bias0.10910.07420.04780.10910.07420.0478−0.00680.3910CP0.91100.91100.91400.92700.94000.93100.85800.7740AL1.91051.90251.89801.63881.65411.63721.66312.0048SD3.38962.73972.98842.42621.96122.13920.63890.7006ρ=0.5M1n = 20Bias0.15800.14650.11760.15800.14650.1176−0.00580.4920CP0.88000.86900.87800.90700.90300.90900.85600.8150AL3.25563.24753.23793.03913.12073.04462.98383.7143SD4.08923.13243.56852.96872.28472.59880.85650.9512n = 40Bias0.11020.08260.05270.11020.08260.0527−0.00460.4445CP0.91800.91200.91400.92900.92700.92900.86300.7860AL2.37292.36592.35962.03302.05762.03552.09052.5668SD4.31273.39143.71843.08792.42922.66290.71660.7921M2n = 20Bias0.15230.13630.09730.15230.13630.0973−0.00580.6213CP0.89600.89200.89100.92100.92400.93100.85600.7880AL3.32183.30803.31023.08773.13273.10672.98383.5378SD4.08693.40043.66952.96762.47382.66720.85650.9286n = 40Bias0.12190.09350.04260.12190.09350.0426−0.00460.6363CP0.92000.92000.92700.93100.93200.93200.86300.7240AL2.42782.41952.41582.08262.09472.08182.09052.4501SD4.28393.60143.81823.06752.57932.73430.71660.7754M3n = 20Bias0.21940.20070.16420.21940.20070.1642−0.00580.6914CP0.87000.85600.86900.89900.89000.90300.85600.7650AL3.20623.19343.19382.98793.05233.00512.98383.5663SD3.99453.16613.52552.90182.30802.56610.85650.9324n = 40Bias0.18560.13610.09110.18560.13610.0911−0.00460.6684CP0.92000.91000.91800.94000.93300.93500.86300.7210AL2.36622.35242.34952.02922.04162.02582.09052.5127SD4.24453.42713.70713.03912.45462.65480.71660.7847ρ=0.8M1n = 20Bias0.22010.20480.16920.22010.20480.16920.00420.8355CP0.84700.84100.84400.87400.87700.88400.84000.7960AL4.23594.20354.20823.94674.01363.95944.07175.0903SD5.38004.16874.68433.91713.05453.42300.99831.1108n = 40Bias0.18130.13830.09160.18130.13830.0916−0.00200.7745CP0.92000.91000.92100.93100.93400.93800.85800.7410AL3.03603.02113.01692.60212.62442.60292.71113.2838SD5.56184.37774.75873.98453.13813.40990.81630.8960M2n = 20Bias0.19140.17900.12460.19140.17900.1246−0.00930.9730CP0.90700.88900.91100.92800.92300.93100.86400.7640AL4.27054.24724.25873.97274.01153.99293.86114.5488SD5.31814.43894.74893.86933.23833.45950.97431.0532n = 40Bias0.15190.13800.06440.15190.13800.0644−0.00200.9899CP0.91500.91600.92000.93300.93100.93900.85800.6760AL3.13343.12013.11832.68762.69982.68782.71113.1629SD5.61444.68274.93364.02283.35663.53540.81630.8809M3n = 20Bias0.28970.25750.20940.28970.25750.2094−0.00931.1480CP0.85000.84900.86400.88300.87100.89200.86400.7380AL4.11244.07604.08903.83513.86623.84303.86114.5670SD5.19164.11054.56243.78323.00983.33250.97431.0546n = 40Bias0.26690.20320.13090.26690.20320.1309−0.00201.1346CP0.90300.90500.91900.92300.92100.93400.85800.6430AL3.04943.01943.02152.61782.61962.60732.71113.1736SD5.54684.41924.77033.97513.16873.41890.81630.8827From Tables3 and 4, we can see the following.(1)
Comparison for different sample sizes: when the sample sizes increase, estimatorsμ^ipw, μ^mi, and μ^aipw have small biases, the coverage probability is big, the AL and the SD are small. So, the finite sample performance of the mean empirical likelihood is as that of the original empirical likelihood(2)
Comparison for different estimation methods: estimatorsμ^ipw, μ^mi, and μ^aipw based on MEL have similar point estimation compared with those based on OEL. In addition, estimators μ^ipw, μ^mi, and μ^aipw based on MEL have bigger CP and smaller SD than those based on OEL. So, the ALs based on MEL are comparable, close to the method based on Y¯(3)
Comparison for different estimating equations: as to the bias, the estimatorμ^aipw is closer than μ^ipw and μ^mi. The CP of the estimator μ^aipw performs better than μ^ipw and μ^mi in many cases. So, the AL of the estimator μ^ipw performs better than μ^aipw and μ^mi in many cases(4)
Comparison for different error situations: the variation range of the bias estimated by OEL and MEL withεi∼N0,1+0.5X2 is much smaller and more stable than the method with εi∼N0,1. But the standard deviation estimated by OEL and MEL with εi∼N0,1+0.5X2 is close to the method with εi∼N0,1(5)
Comparison for different correlation structures: whenρ is large, the biases based on MEL are not stable but are compared to μ^cc. But the CPs and the ALs still perform well, and the ALs are stable. when ρ=0 and ρ=0.2, the MEL method performs better than the OEL methodIn a word, the performances of IPW, MI, and AIPW based on MEL precede those methods based on OEL, and AIPW based on MEL achieves the most accurate results.


## 3.1. Simulation 1: Single Covariate Model

The results and discussion may be presented separately, or in one combined section, and may optionally be divided into headed subsections. Suppose thatgX,θ=ϕlY,μ=Y−μ is the estimating equation for μ. We aim to compare the confidence intervals derived from MEL and OEL, for a given sample size n. We consider different scenarios by generating observations X1,X2,…,Xn from N0,1, respectively.There are four indicators for evaluating method performance. The bias represents the difference between the estimated result and the true value, which measures the accuracy of point estimators. The coverage probability represents the probability that the point estimator falls within the confidence interval of the true value, which measures the stability of point estimators. The average length represents the average length of the confidence interval of the estimated value, and the variance represents the degree of dispersion of the estimated value, which measures the accuracy of interval estimators.Based on 5000 replicates, the bias, coverage proportions, average length, and standard deviation were calculated. The simulation results are presented in Table1.Table 1
Performance of the point estimators and 95% CIs for response mean in simulation 1,εi∼N0,1.
ModelOELMELμ^ipwμ^miμ^aipwY¯μ^ccμ^ipwμ^miμ^aipwY¯μ^ccn = 20Bias−0.01060.01520.00980.00000.0586−0.01060.01520.00980.00000.0586CP0.91150.91300.91550.91350.90850.92900.94700.93750.91350.9085AL1.27771.30511.29991.17711.42631.17641.26071.22121.17711.4263SD1.71541.18951.42980.54470.59781.24150.86091.03480.54470.5978n = 30Bias−0.00980.00880.0030−0.00570.0549−0.00980.00880.0030−0.00570.0549CP0.92850.92850.92950.92100.91250.94100.95300.94750.92100.9125AL1.07181.09311.08570.94701.14620.94330.99840.96800.94701.1462SD1.73301.21931.46890.48870.53671.24510.87601.05530.48870.5367n = 40Bias−0.00790.00760.0020−0.00560.0564−0.00790.00760.0020−0.00560.0564CP0.92950.92850.92650.92650.91250.93800.95300.94450.92650.9125AL0.93370.95010.94340.83891.01140.79890.83580.81400.83891.0114SD1.73191.22831.48130.46030.50491.23950.87911.06010.46030.5049Table1 shows the confidence region characteristics of different empirical likelihood methods when the Xi is the one-dimensional normal distribution with mean 1 and variate 1, denoted as Xi∼N1,1. Yi=Xi+εi with εi∼N0,1 for i=1,2,…,n. Then, we generate δi from the Bernoulli distribution with probability function πXi=1/1+exp−0.6−0.2Xi.From Tables1 and 2, we conclude the following:(1)
Comparison for different sample sizes: when the sample sizes increase, all coverage probability estimated by OEL or MEL increases for small sample sizen and all coverage probability estimated by OEL or MEL decreases. However, the bias estimated by AIPW and MI method reduced significantly, and the bias estimated by the complete-case and IPW method changed a little(2)
Comparison of different estimation methods: MEL is much better than OEL. All coverage probabilities estimated by MEL are closer to the nominal levels than those of the OEL method. All standard deviations by MEL are much smaller than those of the OEL method. All average lengths by MEL are much smaller than those of the OEL method(3)
Comparison for different estimating equations: the point estimator based on MI is superior to other estimating equations. In particular, the coverage probabilities estimated by the MI method perform very well in all simulations. In a word, the sequence about the accuracy of the estimating equation is that the MI method is superior to the AIPW method and the AIPW method is superior to IPW(4)
Comparison for different error situations: the average length estimated by OEL and MEL withεi∼N0,1+0.5X2 is much longer and more stable than the method with εi∼N0,1. But the standard deviation estimated by OEL and MEL with εi∼N0,1+0.5X2 is much bigger than the method with εi∼N0,1. Hence, when the errors in the method are heteroscedastic with εi∼N0,1+0.5X2, MEL method for response mean also plays wellTable 2
Performance of the point estimators and 95% CIs for response mean in simulation 1,εi∼N0,1+0.5X2.
ModelOELMELμ^ipwμ^miμ^aipwY¯μ^ccμ^ipwμ^miμ^aipwY¯μ^ccn = 20Bias−0.01030.01460.00850.00110.0588−0.01030.01460.00850.00110.0588CP0.90800.91550.91000.91750.92100.92800.93900.93650.91750.9210AL1.59121.63471.61771.43391.74871.47721.59651.52631.43391.7487SD2.02561.43451.76410.60030.66051.46621.03851.27700.60030.6605n = 30Bias−0.01010.00770.0025−0.00760.0508−0.01010.00770.0025−0.00760.0508CP0.92350.92750.92450.91450.91400.94000.95450.94400.91450.9140AL1.33171.36841.34921.15241.40581.17971.26511.20831.15241.4058SD2.05501.46871.80800.53850.59351.47641.05521.29900.53850.5935n = 40Bias−0.01370.0024−0.0041−0.00920.0503−0.01370.0024−0.0041−0.00920.0503CP0.92500.93000.92900.92800.91600.93550.95400.94400.92800.9160AL1.16361.19431.17651.01961.24211.00041.06371.01901.01961.2421SD2.05541.47991.82860.50700.55891.47101.05921.30870.50700.5589In a word, the performances of IPW, MI, and AIPW based on MEL precede those methods based on OEL, and AIPW based on MEL achieves the most accurate results.


## 3.2. Simulation 2: Multivariate Covariate Model

In simulation 2,Xi′s are 6-dimensional normal distribution with mean 0 and covariance matrix Γ, denoted by Xi∼N0,Γ, where Γ is a 6×6 symmetric matrix with Γjj=1 and Γjj′=ρj′−j for 1≤j<j′≤6, Yi=Xi1+Xi2+⋯+Xi6+εi with εi∼N0,σ2Xi for i=1,…,n. Two kinds of εi′s are considered:(i)
The first generates homoscedastic errors whereεi and Xi are independent and σ2Xi=1(ii)
The second errors are heteroscedastic withσ2Xi=1+0.5Xi12,i=1,2,…,nWe generateδi from the Bernoulli distribution with probability πXi and consider three choices of πXi:(15)M1.πXi=11+exp−0.6−0.2Xi1+Xi2+Xi3,M2.πXi=11+exp−0.2−0.4expXi1+Xi2+Xi3,M3.πXi=1−exp−exp0.1+0.2Xi1+Xi2+Xi3.The coefficients in the propensity models are chosen so that the unconditional rates of missing data are between 20% and 40%. WhenYi are linear and πXi is logistic linear under M1–M3, so SY and Sδ under M1–M3 are one-dimensional.In addition, we also conduct two simulations with four correlation structures:(i)
Independent covariates, where the correlation coefficient is 0(ii)
Slightly correlated covariates, where the correlation coefficient is 0.2(iii)
Moderately correlated covariates, where the correlation coefficient is 0.5(iv)
Strongly correlated covariates, where the correlation coefficient is 0.8All results are based on 1000 replications and the sample sizes aren = 20 and n = 40. Tables 3 and 4 show the biases and standard deviations (SDs) of the point estimators, the average lengths (ALs), and the coverage probabilities (CPs) of confidence intervals (CIs) at the nominal level of 95%. The CIs based on the mean empirical likelihood ratio LMμ are obtained by IMin equation (15). The CIs based on μ^=Y¯ and μcc are obtained by the normal approximation μ^−1.96σ^se,μ^+1.96σ^se, with standard error σse obtained by the square roots of bootstrap variance estimators based on n/4 bootstrap replications, where n is the size of the sample.Table 3
Performance of the point estimators and 95% CIs for response mean in simulation 2,εi∼N0,1.
ModelOELMELμ^ipwμ^miμ^aipwμ^ipwμ^miμ^aipwY¯μ^ccρ=0M1n = 20Bias0.08080.07350.05690.08080.07350.0569−0.00570.2179CP0.88300.88100.87300.90600.91600.90400.86000.8460AL2.23072.23622.22532.07562.14832.09392.08022.5970SD2.81282.15892.44512.03761.56671.77340.71450.7947n = 40Bias0.08020.05930.04650.08020.05930.04650.00050.2186CP0.92500.92200.92000.93600.93900.93400.87200.8380AL1.60841.60411.59731.37851.39561.37731.39591.7330SD2.87622.29722.51722.05851.64421.80170.58580.6519M2n = 20Bias0.12080.10800.08090.12080.10800.0809−0.00570.3595CP0.89600.88100.88600.92100.91200.91400.86000.7970AL2.26992.26692.26382.11302.15982.12652.08022.4904SD2.78702.27312.49812.01901.64881.81120.71450.7792n = 40Bias0.09040.06130.03340.09030.06130.03340.00050.3539CP0.93600.93500.92900.94600.94600.94200.87200.7700AL1.63111.62681.62211.39901.41021.39781.39591.6898SD2.85352.39502.56212.04231.71421.83380.58580.6434M3n = 20Bias0.11510.10580.08270.11510.10580.0827−0.00570.3179CP0.88800.88800.88100.90800.91100.90900.86000.8320AL2.23042.23182.22332.07552.13592.08972.08022.5149SD2.77512.18712.44852.01041.58671.77540.71450.7820n = 40Bias0.10630.07770.05740.10630.07770.05740.00050.3119CP0.91700.92000.92000.93400.93500.93500.87200.7810AL1.60661.60151.59591.37741.39131.37561.39591.6988SD2.84702.31502.51812.03761.65701.80230.58580.6447ρ=0.2M1n = 20Bias0.11540.10440.08410.11540.10440.08410.00540.3065CP0.87100.87900.87000.89900.90700.91100.86700.8330AL2.55722.54942.54342.38282.44502.39432.31662.9036SD3.21472.48592.80352.33031.80762.03660.75360.8401n = 40Bias0.08920.06460.04810.08920.06460.0481−0.00310.2807CP0.91400.90800.91200.93300.92500.92500.85300.8190AL1.83631.83341.82591.57281.59481.57421.58252.0088SD3.31332.62782.87882.37161.88112.06060.62380.7026M2n = 20Bias0.17130.15920.12470.17130.15920.12470.00540.4961CP0.89800.89900.90200.92000.92300.93000.86700.7940AL2.57562.57362.57062.39662.44662.41432.31662.7918SD3.16572.61162.83772.29431.89532.05840.75360.8257n = 40Bias0.09690.06860.03300.09690.06860.0330−0.00310.4320CP0.92700.92400.93000.93900.94500.94300.85300.7840AL1.86301.85991.85461.59741.61141.59781.58251.9375SD3.27852.75022.93052.34671.96862.09760.62380.6895M3n = 20Bias0.16170.14410.11800.16170.14410.11800.00540.4538CP0.87700.87300.87100.90500.90200.90500.86700.8020AL2.56282.55782.55362.38682.44662.40152.31662.8692SD3.19982.51642.81252.31961.82842.04160.75360.8361n = 40Bias0.13570.09980.07310.13570.09980.0731−0.00310.4151CP0.91300.91100.91400.92100.92900.93100.85300.7560AL1.83261.82681.82181.57091.58591.57051.58251.9471SD3.26522.65532.87392.33721.90072.05710.62380.6908ρ=0.5M1n = 20Bias0.15350.14110.11490.15350.14110.11490.00890.5182CP0.87100.87100.86900.90400.89600.90300.86000.8020AL3.21163.19633.19492.99263.05613.00662.95223.6533SD4.06803.15483.52972.95272.29922.56860.85110.9427n = 40Bias0.14130.10510.07870.14130.10510.0787−0.00120.4787CP0.90100.90700.90600.91600.92500.92100.85400.7730AL2.30342.29672.28961.97291.99551.97402.00392.5153SD4.20033.32003.61193.00742.37782.58650.70180.7849M2n = 20Bias0.20560.19010.14810.20560.19010.14810.00890.7143CP0.89500.89500.89900.92200.92000.92800.86000.7810AL3.26693.25753.25753.03653.08993.05462.95223.5512SD4.01943.34943.60822.91772.43592.62160.85110.9314n = 40Bias0.12600.09650.04690.12600.09650.0469−0.00120.6236CP0.93300.93000.93600.94300.94000.94200.85400.7430AL2.36122.35692.35152.02442.03982.02582.00392.4226SD4.17233.51643.71872.98722.51792.66250.70180.7707M3n = 20Bias0.31240.28020.22890.31240.28020.22890.01511.2031CP0.87300.85100.87500.90500.88300.90600.86900.7220AL4.14184.09184.11093.86003.87673.85643.86034.6516SD5.21214.15754.57683.79323.03983.33800.97351.0647n = 40Bias0.19270.13760.09250.19270.13760.0925−0.00120.6938CP0.91200.90700.91700.92200.92300.92900.85400.6960AL2.31482.30152.29771.98481.99651.98082.00392.4544SD4.17193.36483.62802.98722.40982.59790.70180.7748ρ=0.8M1n = 20Bias0.22080.21080.17870.22080.21080.17870.01510.8554CP0.86300.85100.86300.89600.89400.90300.86900.7790AL4.14784.11974.12333.86813.93673.87693.86034.7258SD5.26784.10584.58753.83293.00623.35010.97351.0731n = 40Bias0.20780.16580.12020.20780.16580.12020.00270.8162CP0.90000.90600.91300.92000.91900.92500.85600.7350AL2.98512.97302.96722.55762.58212.55932.60543.2279SD5.48544.31724.68553.93033.09543.35810.80000.8886M2n = 20Bias0.24860.23890.18060.24860.23890.18060.01511.0812CP0.89700.88100.89700.92400.90900.92200.86900.7530AL4.24644.21054.21973.95233.97853.96073.86034.5830SD5.27694.41564.71843.84043.22283.43850.97351.0572n = 40Bias0.16110.14310.07090.16110.14310.07090.00270.9679CP0.93000.92800.93300.94700.93600.95000.85600.6910AL3.07433.06443.06062.63222.65012.63642.60543.0973SD5.48674.61034.84603.93073.30393.47190.80000.8721M3n = 20Bias0.31240.28020.22890.31240.28020.22890.01511.2031CP0.87300.85100.87500.90500.88300.90600.86900.7220AL4.14184.09184.11093.86003.87673.85643.86034.6516SD5.21214.15754.57683.79323.03983.33800.97351.0647n = 40Bias0.29100.21930.14720.29100.21930.14720.00271.1758CP0.90400.90000.91100.92000.92200.93700.85600.6100AL3.00012.97232.97382.57392.57682.56442.60543.1358SD5.45744.36974.69883.91113.13283.36710.80000.8764Table 4
Performance of the point estimators and 95% CIs for response mean in simulation 2,εi∼N0,1+0.5Xi12.
ModelOELMELμ^ipwμ^miμ^aipwμ^ipwμ^miμ^aipwY¯μ^ccρ=0M1n = 20Bias0.09800.09550.08040.09800.09550.0804−0.00530.2177CP0.89000.88100.88400.91000.91200.91500.85300.8350AL2.35742.36342.35402.19892.27972.21962.15192.6703SD2.94162.23822.57612.13151.62581.86950.72640.8057n = 40Bias0.06060.04010.02680.06060.04010.02680.00190.2058CP0.92300.92100.91900.93300.93700.93300.86800.8320AL1.67241.66901.66131.43371.45391.43281.44101.8057SD2.97632.36702.61562.13021.69421.87210.59480.6646M2n = 20Bias0.12270.11170.08560.12270.11170.0856−0.00530.3586CP0.87900.87900.88100.90200.91500.90600.85300.8150AL2.35492.35682.34872.19402.24952.20702.15192.5743SD2.88802.34792.58982.09231.70411.87870.72640.7921n = 80Bias0.08110.05080.02430.08110.05080.02430.00190.3421CP0.93600.93100.93500.94400.94600.94700.86800.7940AL1.69701.69321.68761.45641.46971.45491.44101.7536SD2.95642.47052.66112.11591.76821.90460.59480.6549M3n = 20Bias0.13920.12660.10500.13920.12660.1050−0.00530.3263CP0.87800.87900.88100.91100.91500.91600.85300.8290AL2.34072.34312.33612.18202.24602.19912.15192.5981SD2.88942.26872.56502.09321.64661.86040.72640.7947n = 80Bias0.09180.06500.04370.09180.06500.04370.00190.3066CP0.92900.92800.93200.93900.94600.94300.86800.8030AL1.67501.66841.66291.43721.45161.43441.44101.7434SD2.95892.38672.61832.11771.70831.87400.59480.6531ρ=0.2M1n = 20Bias0.09910.09800.07890.09910.09800.0789−0.00570.2874CP0.87500.87300.87700.89700.91500.90400.85800.8330AL2.63302.63992.62972.45512.54502.47932.38122.9765SD3.30562.50422.87982.39651.82042.09150.76510.8508n = 40Bias0.06660.04520.02620.06660.04520.0262−0.00680.2749CP0.90200.91500.91100.91000.93500.92900.85800.8030AL1.90211.89721.89121.62991.65171.63161.66312.0389SD3.42672.69802.97572.45271.93132.13000.63890.7060M2n = 20Bias0.12650.11330.08430.12650.11330.0843−0.00570.4260CP0.89500.90000.90000.91900.92700.92500.85800.7880AL2.66132.65892.65222.47872.53092.49082.38122.8391SD3.25972.68072.93232.36361.94682.12830.76510.8321n = 40Bias0.08920.05670.02440.08920.05670.0244−0.00680.4145CP0.92300.92300.91900.93400.93500.93600.85800.7650AL1.93391.92891.92461.65911.67221.65901.66311.9615SD3.38372.83913.03652.42202.03232.17350.63890.6934M3n = 20Bias0.16460.14770.12370.16460.14770.1237−0.00570.4220CP0.88300.87300.87900.90200.91000.90700.85800.8010AL2.59712.59842.58882.42252.48592.43142.38122.8912SD3.20572.54332.84972.32431.84812.06900.76510.8394n = 40Bias0.10910.07420.04780.10910.07420.0478−0.00680.3910CP0.91100.91100.91400.92700.94000.93100.85800.7740AL1.91051.90251.89801.63881.65411.63721.66312.0048SD3.38962.73972.98842.42621.96122.13920.63890.7006ρ=0.5M1n = 20Bias0.15800.14650.11760.15800.14650.1176−0.00580.4920CP0.88000.86900.87800.90700.90300.90900.85600.8150AL3.25563.24753.23793.03913.12073.04462.98383.7143SD4.08923.13243.56852.96872.28472.59880.85650.9512n = 40Bias0.11020.08260.05270.11020.08260.0527−0.00460.4445CP0.91800.91200.91400.92900.92700.92900.86300.7860AL2.37292.36592.35962.03302.05762.03552.09052.5668SD4.31273.39143.71843.08792.42922.66290.71660.7921M2n = 20Bias0.15230.13630.09730.15230.13630.0973−0.00580.6213CP0.89600.89200.89100.92100.92400.93100.85600.7880AL3.32183.30803.31023.08773.13273.10672.98383.5378SD4.08693.40043.66952.96762.47382.66720.85650.9286n = 40Bias0.12190.09350.04260.12190.09350.0426−0.00460.6363CP0.92000.92000.92700.93100.93200.93200.86300.7240AL2.42782.41952.41582.08262.09472.08182.09052.4501SD4.28393.60143.81823.06752.57932.73430.71660.7754M3n = 20Bias0.21940.20070.16420.21940.20070.1642−0.00580.6914CP0.87000.85600.86900.89900.89000.90300.85600.7650AL3.20623.19343.19382.98793.05233.00512.98383.5663SD3.99453.16613.52552.90182.30802.56610.85650.9324n = 40Bias0.18560.13610.09110.18560.13610.0911−0.00460.6684CP0.92000.91000.91800.94000.93300.93500.86300.7210AL2.36622.35242.34952.02922.04162.02582.09052.5127SD4.24453.42713.70713.03912.45462.65480.71660.7847ρ=0.8M1n = 20Bias0.22010.20480.16920.22010.20480.16920.00420.8355CP0.84700.84100.84400.87400.87700.88400.84000.7960AL4.23594.20354.20823.94674.01363.95944.07175.0903SD5.38004.16874.68433.91713.05453.42300.99831.1108n = 40Bias0.18130.13830.09160.18130.13830.0916−0.00200.7745CP0.92000.91000.92100.93100.93400.93800.85800.7410AL3.03603.02113.01692.60212.62442.60292.71113.2838SD5.56184.37774.75873.98453.13813.40990.81630.8960M2n = 20Bias0.19140.17900.12460.19140.17900.1246−0.00930.9730CP0.90700.88900.91100.92800.92300.93100.86400.7640AL4.27054.24724.25873.97274.01153.99293.86114.5488SD5.31814.43894.74893.86933.23833.45950.97431.0532n = 40Bias0.15190.13800.06440.15190.13800.0644−0.00200.9899CP0.91500.91600.92000.93300.93100.93900.85800.6760AL3.13343.12013.11832.68762.69982.68782.71113.1629SD5.61444.68274.93364.02283.35663.53540.81630.8809M3n = 20Bias0.28970.25750.20940.28970.25750.2094−0.00931.1480CP0.85000.84900.86400.88300.87100.89200.86400.7380AL4.11244.07604.08903.83513.86623.84303.86114.5670SD5.19164.11054.56243.78323.00983.33250.97431.0546n = 40Bias0.26690.20320.13090.26690.20320.1309−0.00201.1346CP0.90300.90500.91900.92300.92100.93400.85800.6430AL3.04943.01943.02152.61782.61962.60732.71113.1736SD5.54684.41924.77033.97513.16873.41890.81630.8827From Tables3 and 4, we can see the following.(1)
Comparison for different sample sizes: when the sample sizes increase, estimatorsμ^ipw, μ^mi, and μ^aipw have small biases, the coverage probability is big, the AL and the SD are small. So, the finite sample performance of the mean empirical likelihood is as that of the original empirical likelihood(2)
Comparison for different estimation methods: estimatorsμ^ipw, μ^mi, and μ^aipw based on MEL have similar point estimation compared with those based on OEL. In addition, estimators μ^ipw, μ^mi, and μ^aipw based on MEL have bigger CP and smaller SD than those based on OEL. So, the ALs based on MEL are comparable, close to the method based on Y¯(3)
Comparison for different estimating equations: as to the bias, the estimatorμ^aipw is closer than μ^ipw and μ^mi. The CP of the estimator μ^aipw performs better than μ^ipw and μ^mi in many cases. So, the AL of the estimator μ^ipw performs better than μ^aipw and μ^mi in many cases(4)
Comparison for different error situations: the variation range of the bias estimated by OEL and MEL withεi∼N0,1+0.5X2 is much smaller and more stable than the method with εi∼N0,1. But the standard deviation estimated by OEL and MEL with εi∼N0,1+0.5X2 is close to the method with εi∼N0,1(5)
Comparison for different correlation structures: whenρ is large, the biases based on MEL are not stable but are compared to μ^cc. But the CPs and the ALs still perform well, and the ALs are stable. when ρ=0 and ρ=0.2, the MEL method performs better than the OEL methodIn a word, the performances of IPW, MI, and AIPW based on MEL precede those methods based on OEL, and AIPW based on MEL achieves the most accurate results.


## 4. Real Data Analysis

In this section, we apply the proposed method to the Boston Housing data to illustrate our proposed MEL for missing data. The data is taken from the UCI Irvine Machine Learning Repository. The distribution of the per capita crime rate (CRIM) by the town is unknown. The inference on CRIM distribution based on the full data has even been analyzed by Liang et al. [15]. The dataset consists of 506 observations and 14 variates, with CRIM being the response variate and the other 11 being the covariates. We are interested in the mean of CRIM and its confidence region. We set the unconditional rates of missing data between 5% and 15%, and all real data analysis results are based on 1000 replications and the sample size is n = 20, n = 40, and n = 80. Tables 5–7 show the biases and standard deviations (SDs) of the point estimators, the average lengths (ALs), and the coverage probabilities (CPs) of confidence intervals (CIs) at the nominal level 95%.Table 5
Performance of estimation for response mean in the Boston data (rep = 1000,b = 20, n = 20).
Bias methodIPW3.537MI3.548AIPW3.538Y: 3.609CC: 3.899MELOELMELOELMELOELCP0.8850.8550.880.8550.8870.8590.7730.807AL6.29036.61736.38416.61966.33626.65015.82526.3198SD5.61047.66125.10666.95495.48567.48601.15511.2012Table 6
Performance of estimation for response mean in the Boston data (rep = 1000,b = 20, n = 40).
Bias methodIPW3.501MI3.501AIPW3.504Y: 3.5898CC: 3.8859MELOELMELOELMELOELCP0.8910.8670.8930.8670.8920.8680.8430.817AL4.43304.94114.43814.94294.43494.94194.82704.4573SD5.47187.61695.43257.56195.45187.58891.07021.0291Table 7
Performance of estimation for response mean in the Boston data (rep = 1000,b = 20, n = 80).
Bias MethodIPW3.595MI3.600AIPW3.599Y: 3.649CC: 3.956MELOELMELOELMELOELCP0.9180.9040.9180.9050.9180.9030.850.88AL3.17443.78683.17853.78893.17663.78813.49333.7886SD5.83558.19445.79438.13655.81588.16660.92160.9600In Tables5–7, it is observed that when the sample sizes increase, the coverage probabilities of the estimators μ^ipw, μ^mi, and μ^aipw based on the MEL method are bigger than those of the AL and the SD. In particular, the confidence regions of the response mean are close to the confidence region of the full data. So, the finite sample performance of mean empirical likelihood is better than that of the original empirical likelihood.


## 5. Conclusions

This paper proposed a new empirical likelihood for inference on response mean with a small sample and multidimensional variate using the mean empirical likelihood. Its large sample properties were proved with different bias-corrected estimating equations. This new method outperforms the original empirical likelihood methods.On the basis of IPW, MI, and AIPW, the new method uses a pairwise-mean dataset to obtain its advantages. Compared with the original EL method, this method can obtain an approximate point estimator and better CPs, ALs, and SDs. Compared with the other EL inference methods, this method has two characteristics: one is that the size of the sample is much larger than dimensions, and the other is that the number of samples is large. In addition, AIPW based on MEL achieves the most accurate results. Theoretically, when the dimension of the covariate is increasing, the performance of response mean inference based EL is bad. The method of Liang et al. [15] can solve this problem. Meanwhile, it has superior performance when the size of the sample is small. Therefore, we recommend that three bias-corrected models be carefully constructed and used to infer the confidence region of response mean with missing at random. We will further study the MEL method for categorical data with random missing covariates.



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*Source: 8893594-2020-12-15.xml*