The Estimation of Variances After Using a Gaussianating Transformation

Abstract

Neyman and Scott [12] considered the problem of estimating the mean of a distribution after some fixed Gaussian inducing transformation had been applied to the observations. More specifically, if a random variable $x$ is observed then it is assumed that $\xi = f{}^{-1}(x)$ has a Gaussian distribution with mean $\mu$ and variance $\sigma^2$. They find the minimum variance unbiased estimator (MVUE), $\hat\theta$, of $\hat{\theta} = E(x)$ in terms of the MVUE's of $\mu$ and $\sigma^2$. The heart of their solution is in taking a Taylor series expansion of $f(\xi)$ around the origin and showing that the resulting infinite power series behaves like a finite one in the sense that the operations of taking expectations and summing can be reversed as required provided only that $f(\xi)$ is an entire function of second order or less. This paper exploits the Taylor series expansion of $f(\xi)$ around the mean to find the MVUE of $\phi^2 = E(x - \theta)^2$. It was motivated in part by the fact that recent research in cloud seeding has shown that the variance may be a more important parameter than the mean [8], [15], and also by a general concern with the class of recursive transformations defined on p. 651 of [12]. In addition the MVUE of Var $(\hat{\theta})$ for particular transformations is derived. The problems under consideration here have been considered by Finney [6], Sichel [14], and Meulenberg [11] for the logarithmic transformation i.e., $f(\xi) = e^{m\xi}$. Their results provide a check on mine. Schmetterer [13] has interpreted the results of Neyman and Scott in terms of the solution $h(\hat{\mu}, \hat{\sigma}^2)$ of the integral equation $E\lbrack h(\hat{\mu}, \hat{\sigma}^2)\rbrack = E(x)$ where $\hat{\mu}, \hat{\sigma}^2$ are the MVUE's for $\mu$ and $\sigma^2$. Kolmogorov [10] has also considered the problem of finding unbiased estimators in terms of the solutions of integral equations but he relies heavily upon the results of Blackwell [2] in using the sufficient statistics for $(\mu, \sigma^2)$ to turn unbiased but inefficient estimators into the MVUE. The problem discussed here can be formulated as an integral equation: viz., find $h\ast(\hat{\mu}, \hat{\sigma}^2)$ such that $E\lbrack h^\ast(\hat{\mu}, \hat{\sigma}^2_\rho)\rbrack = E\lbrack x - E(x)\rbrack^2$ but the present author has not attempted to solve this problem in this way. Rather he has approached the problem by a straightforward application of the method of Neyman and Scott.