Unbiased estimation in one-parameter exponential families for the inverse of the natural parameter with extensions

TL;DR

This paper develops unbiased estimators for the inverse of the natural parameter in one-parameter exponential families, extending previous results and applying to various distributions and functions.

Contribution

It introduces new unbiased estimators for inverse parameters in exponential families, including extensions to negative powers and complete monotone functions.

Findings

Unbiased estimator for 1/θ in exponential families when θ > 0

Applications to Gamma, Inverse Gaussian, and truncated distributions

Extensions to estimating θ^{-k} and complete monotone functions

Abstract

For one-parameter continuous exponential families, we identify an unbiased estimator of the inverse of the natural parameter $\theta$ for cases where $\theta > 0$, extending an earlier result of \cite{voinov1985unbiased} applicable to a normal model. We provide various applications for Gamma models, Inverse Gaussian models, distributions obtained by truncation, and ratios of normal means. Moreover, we extend the findings to estimating negative powers $\theta^{-k}$, and more generally to complete monotone functions $q(\theta)$.