Location and Scale Parameter Family of Distributions Attaining the Bhattacharyya Bound

Abstract

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.

Keywords:

• Bhattacharyya inequality
• Differential equation
• Exponential family
• Location and scale parameter family

Mathematics Subject Classification:

• 62F10
• 62E15

Acknowledgments

The author would like to thank the referees and the editor for carefully reading the paper and for their useful comments in improving it. The author would like to express to Professor M. Akahira of the University of Tsukuba his deepest gratitude for the comments offered. The author would also like to thank Dr. T. Ohnishi of the Institute of Statistical Mathematics for his useful comment. This research was supported by Grant-in-Aid for Scientific Research, Young Scientists (B) (No. 15700229), Japan Society for the Promotion of Science.

Notes

Present address: Graduate School of Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka 599-8531, Japan; E-mail: [email protected]