Biased-sample empirical likelihood weighting: an alternative to inverse probability weighting

TL;DR

This paper introduces a biased-sample empirical likelihood weighting (ELW) method as a stable and efficient alternative to inverse probability weighting (IPW) for handling unrepresentative, missing, or biased data, overcoming IPW's instability issues.

Contribution

The paper proposes the ELW method that avoids inverse probabilities, ensuring stability and improved efficiency over traditional IPW estimators in various sampling and missing data scenarios.

Findings

ELW weights are always well defined and easy to implement.

ELW estimator is asymptotically normal and more efficient than IPW.

ELW outperforms IPW in mean square error in simulations and real data.

Abstract

Inverse probability weighting (IPW) is widely used in many areas when data are subject to unrepresentativeness, missingness, or selection bias. An inevitable challenge with the use of IPW is that the IPW estimator can be remarkably unstable if some probabilities are very close to zero. To overcome this problem, at least three remedies have been developed in the literature: stabilizing, thresholding, and trimming. However the final estimators are still IPW type estimators, and inevitably inherit certain weaknesses of the naive IPW estimator: they may still be unstable or biased. We propose a biased-sample empirical likelihood weighting (ELW) method to serve the same general purpose as IPW, while completely overcoming the instability of IPW-type estimators by circumventing the use of inverse probabilities. The ELW weights are always well defined and easy to implement. We show theoretically that the ELW estimator is asymptotically normal and more efficient than the IPW estimator and its stabilized version for missing data problems and unequal probability sampling without replacement. Its asymptotic normality is also established under unequal probability sampling with replacement. Our simulation results and a real data analysis indicate that the ELW estimator is shift-equivariant, nearly unbiased, and usually outperforms the IPW-type estimators in terms of mean square error.