On the Attainment of Cramer-Rao and Bhattacharyya Bounds for the Variance of an Estimate

Abstract

If a variable $X$ has density function $f(x, \theta)$, then in many cases the Cramer-Rao bound or the Bhattacharyya bounds may be used to show that a function $d(x)$ is a uniformly minimum variance unbiased estimate of the real parameter $\theta$. In this paper it is shown that if $f(x, \theta)$ is a member of the family of densities of the Darmois-Koopman form, and if the variance of $d(x)$ achieves the $k$th Bhattacharyya bound, but not the $(k - 1)$th bound, then $f(x, \theta) = \exp\lbrack t(x)g(\theta) + g_0(\theta) + h(x)\rbrack$ and $d(x)$ is a polynomial in $t(x)$ of degree $k$. Further, the variance of any polynomial in $t(x)$ of degree $k$ will achieve the $k$th bound, so that if any such unbiased polynomial exists, it will necessarily be uniformly minimum variance unbiased. Some properties of these polynomial estimates are discussed.