Estimating Generating Functions

Abstract

This note shows that, under appropriate conditions, if a function $A(\theta; t)$ of an unknown parameter $\theta$ and a real variable $t$ has an infinite series expansion and if there is a function $B(S; t)$ of the sufficient statistic $S$ which is an unbiased estimator of $A$ for every $t$ and which also has an infinite series expansion, then the coefficients of the power of $t$ in the expansion of $B$ are the proper estimators for the coefficients of the corresponding powers in the expansion of $A$. This result is applied to estimate two functions of the normal parameters, $\mu$ and $\sigma^2$, which arise in the derivation of expressions for the removal of transformation bias.