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.
翻译:当数据不具有代表性、缺失或选择偏差时,在许多领域广泛使用反正概率加权法(IPW),当数据不具有代表性、缺失或选择偏差时,在许多领域广泛使用。使用IPW的一个不可避免的挑战是,如果某些概率非常接近于零,IPW的估算器可能非常不稳定。为了克服这一问题,文献中至少已经开发了三种补救措施:稳定、阈值和裁剪。然而,最后的估算器仍然是IPW 类型估计器,并且不可避免地继承天真的 IPW 估算器的某些弱点:它们可能仍然不稳定或偏差。我们建议一种偏差缩缩略经验概率加权法(ELW),用于与 IPW 一样的一般目的,而如果某些概率非常接近,则完全克服 IPW 类估算器的不稳定性。 ELW 加权法总是很明确和易于执行。 我们从理论上看, ELW 估算器比 IPW 估测算器的正常正常正常和更有效率: IPW 估测器及其稳定版本的经验概率加权加权加权加权加权的加权比 。