On a Relation Between a Family of Distributions Attaining the Bhattacharyya Bound and That of Linear Combinations of the Distributions from an Exponential Family

Abstract

An unbiased estimation problem of a function g(θ) of a real parameter is considered. A relation between a family of distributions for which an unbiased estimator of a function g(θ) attains the general order Bhattacharyya lower bound and that of linear combinations of the distributions from an exponential family is discussed. An example on a family of distributions involving an exponential and a double exponential distributions with a scale parameter is given. An example on a normal distribution with a location parameter is also given.

Keywords:

• Bhattacharyya bound
• Exponential family
• Linear differential equation

Acknowledgments

The author wishes to thank Professor M. Akahira of the University of Tsukuba for giving helpful advice and the referee for the kind comments.