Estimating the reciprocal of a binomial proportion

TL;DR

This paper introduces an optimal shrinkage estimator for the reciprocal of a binomial proportion, addressing the zero-event problem and demonstrating improved performance over existing methods through simulations and real data application.

Contribution

It develops a new optimal shrinkage estimator for the inverse binomial proportion and derives explicit formulas for the shrinkage parameter, enhancing estimation accuracy.

Findings

The new estimator outperforms existing methods in most practical scenarios.

Simulation results confirm the estimator's superior or comparable performance.

Application to COVID-19 data illustrates practical usefulness.

Abstract

As a classic parameter from the binomial distribution, the binomial proportion has been well studied in the literature owing to its wide range of applications. In contrast, the reciprocal of the binomial proportion, also known as the inverse proportion, is often overlooked, even though it also plays an important role in various fields including clinical studies and random sampling. The maximum likelihood estimator of the inverse proportion suffers from the zero-event problem, and to overcome it, alternative methods have been developed in the literature. Nevertheless, there is little work addressing the optimality of the existing estimators, as well as their practical performance comparison. Inspired by this, we propose to further advance the literature by developing an optimal estimator for the inverse proportion in a family of shrinkage estimators. We further derive the explicit and approximate formulas for the optimal shrinkage parameter under different settings. Simulation studies show that the performance of our new estimator performs better than, or as well as, the existing competitors in most practical settings. Finally, to illustrate the usefulness of our new method, we also revisit a recent meta-analysis on COVID-19 data for assessing the relative risks of physical distancing on the infection of coronavirus, in which six out of seven studies encounter the zero-event problem.