# Unbiased Estimation of the Reciprocal Mean for Non-negative Random Variables

## TL;DR

This paper analyzes unbiased Monte Carlo estimators for the reciprocal of the expectation of a non-negative random variable, focusing on their theoretical properties, optimal parameter choices, and practical confidence intervals.

## Contribution

It provides a detailed theoretical investigation of the properties of unbiased estimators for 1/ E[Z], including optimal parameter selection and asymptotic behavior.

## Key findings

- Expected time variance product decreases as w decreases.
- Estimator is asymptotically equivalent to the maximum likelihood biased ratio estimator.
- Practical confidence intervals are established.

## Abstract

Many simulation problems require the estimation of a ratio of two expectations. In recent years Monte Carlo estimators have been proposed that can estimate such ratios without bias. We investigate the theoretical properties of such estimators for the estimation of $\beta = 1/\mathbb{E}\, Z$, where $Z \geq 0$. The estimator, $\widehat \beta(w)$, is of the form $w/f_w(N) \prod_{i=1}^N (1 - w\, Z_i)$, where $w < 2\beta$ and $N$ is any random variable with probability mass function $f_w$ on the positive integers. For a fixed $w$, the optimal choice for $f_w$ is well understood, but less so the choice of $w$. We study the properties of $\widehat \beta(w)$ as a function of~$w$ and show that its expected time variance product decreases as $w$ decreases, even though the cost of constructing the estimator increases with $w$. We also show that the estimator is asymptotically equivalent to the maximum likelihood (biased) ratio estimator and establish practical confidence intervals.