Location and Scale Parameter Family of Distributions Attaining the Bhattacharyya Bound

It is well known that, under appropriate regularity conditions, the variance of an unbiased estimator of a real-valued function of an unknown parameter can coincide with the Cramér–Rao lower bound only if the family of distributions is a one-parameter exponential family. But it seems that the necessary conditions about the probability distribution for which there exists an unbiased estimator whose variance coincides with the Bhattacharyya lower bound are not completely known. The purpose of this paper is to specify the location, scale, and location-scale parameter family of distributions attaining the general order Bhattacharyya bound in certain class.