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, θ), then in manycases the Cramer-Rao bound or the Bhattacharyya bounds may be usedto show that a function d(x) is a uniformly minimum variance unbiasedestimate of the real parameter θ. In this paper it is shownthat if f(x, θ) is a member of the family of densities ofthe Darmois-Koopman form, and if the variance of d(x) achievesthe kth Bhattacharyya bound, but not the (k - 1)th bound, thenf(x, θ) = \exp\lbrack t(x)g(θ) + g_0(θ) + h(x)\rbrackand d(x) is a polynomial in t(x) of degree k. Further, thevariance of any polynomial in t(x) of degree k will achieve thekth bound, so that if any such unbiased polynomial exists, it willnecessarily be uniformly minimum variance unbiased. Some propertiesof these polynomial estimates are discussed.