Uniform improvement of empirical likelihood for missing response problem

Kwun Chuen Gary Chan

doi:10.1214/12-ejs673

Abstract

An empirical likelihood (EL) estimator was proposed by Qin and Zhang (2007) for improving the inverse probability weighting estimation in a missing response problem. The authors showed by simulation studies that the finite sample performance of EL estimator is better than certain existing estimators and they also showed large sample results for the estimator. However, the empirical likelihood estimator does not have a uniformly smaller asymptotic variance than other existing estimators in general. We consider several modifications to the empirical likelihood estimator and show that the proposed estimator dominates the empirical likelihood estimator and several other existing estimators in terms of asymptotic efficiencies under missing at random. The proposed estimator also attains the minimum asymptotic variance among estimators having influence functions in a certain class and enjoys certain double robustness properties.

Highlights

Introduction and existing estimatorsSuppose we are interested in estimating the mean μ of a random variable Y butY is partially observed subject to missingness
The main purpose of this paper is to study modifications of the empirical likelihood estimator that attains the minimum asymptotic variance among class L, and theoretically dominate the EL estimator and many other existing estimators when the same amount of covariate information is used
Before we show that empirical likelihood attains the minimum efficiency among class L′, we first characterize the conditions for m(X) to achieve minimum efficiency

Summary

Introduction and existing estimators

Y is partially observed subject to missingness. Let X be a vector of covariates that are fully observable and R be an indicator that Y is observed. An empirical likelihood estimator is proposed by Qin and Zhang (2007) defined as n μEL =. Following Qin and Lawless (1994), Qin, Zhang and Leung (2009) studied a single-step empirical likelihood estimator for missing data problems. The empirical likelihood estimator has nice small sample properties shown in simulations, but it does not theoretically dominate other existing estimators in terms of asymptotic efficiency for arbitrary pre-specified a(X). The main purpose of this paper is to study modifications of the empirical likelihood estimator that attains the minimum asymptotic variance among class L, and theoretically dominate.

Optimality of empirical likelihood under missing completely at random

Simulations

Discussions