An estimator for Poisson means whose relative error distribution is known

TL;DR

This paper introduces a new estimator for Poisson means that has a known, distribution-independent relative error, enabling exact confidence intervals and efficient high-dimensional integration, especially useful in Monte Carlo methods.

Contribution

The paper presents a novel Poisson mean estimator with a known relative error distribution, facilitating exact confidence intervals and applications in high-dimensional Monte Carlo integration.

Findings

The estimator's relative error distribution is independent of the true mean.

It allows for exact confidence intervals for Poisson means.

Application demonstrated in estimating the Ising model's normalizing constant.

Abstract

Suppose that $X_1,X_2,\ldots$ are a stream of independent, identically distributed Poisson random variables with mean $\mu$. This work presents a new estimate $\mu_k$ for $\mu$ with the property that the distribution of the relative error in the estimate ($(\hat \mu_k/\mu) - 1$) is known, and does not depend on $\mu$ in any way. This enables the construction of simple exact confidence intervals for the estimate, as well as a means of obtaining fast approximation algorithms for high dimensional integration using TPA. The new estimate requires a random number of Poisson draws, and so is best suited to Monte Carlo applications. As an example of such an application, the method is applied to obtain an exact confidence interval for the normalizing constant of the Ising model.