Bhattacharyya and Kshirsagar Bounds in Generalized Gamma Distribution

Abstract

The generalized gamma (GG) distribution, which was described for the first time by Stacy (1962), is a popular distribution in many areas such as reliability, modeling data, actuarial studies, and applied statistics, because it is extremely flexible. The estimation of its parameters has gathered the attention of many authors such as Manning et al. (2002), Huang and Hwang (2006), and Gomes et al. (2008). In this article, we focus on the accuracy of the estimators of the parameter functions of the GG distribution, based on finding the best lower bound for the variance of the unbiased estimators. In order to do this, we obtain the general form of Bhattacharyya matrix and compare the two famous lower bounds for the variance, which are the Bhattacharyya and Kshirsagar bounds for variances of different estimators of the parameter functions such as reliability, hazard rate, mode, and median. Then we propose the suitable bounds in each case.

Keywords:

• Bhattacharyya bound
• Cramer–Rao bound
• Generalized gamma distribution
• Hammersley–Chapman–Robbins bound
• Hazard rate
• Kshirsagar bound
• Reliability function

Mathematics Subject Classification:

• 65D20
• 62N05
• 62F99
• 62P30

Acknowledgments

The authors are grateful to the referees for their useful comments and suggestions. Also, the support of Ordered and Spatial Data Center of Excellence of Ferdowsi University of Mashhad is acknowledged.