A Zero-Modified Generalized Negative Binomial Distribution Based on Stacy's Generalized Gamma

• Activity Number: 12
• Type: Topic Contributed
• Date/Time: Sunday, July 29, 2012 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Ram Tripathi*+ and Ramesh Gupta
• Companies: The University of Texas at San Antonio and University of Maine
• Keywords: Negative Binomial ; Stacy's gamma ; zero-modified distribution ; maximum likelihood estimation ; moment estimation ; relative error

Abstract

The negative binomial distribution has been found suitable for describing a variety of data in such areas as biology, entomology, ecology and accident statistics. Various generalizations of this model are available in the literature. The authors have developed a generalized negative binomial (STGNBD) based on Stacy's generalized gamma. Often in practice, one encounters data which have more or fewer zeros than predicted by the negative binomial or its generalizations. In this paper, we develop a zero-modified version of STGNBD. The expressions for its mean and variance are derived. The moment type estimators as well as the maximum likelihood estimators for its parameters are developed. The score test is developed for testing if the STGNBD is an appropriate model rather than zero-modified STGNBD proposed here. This model is fitted to some data sets from the literature along with the fits afforded by other similar models from the literature. Graphical comparisons are made between the shapes of the STGNBD with the shapes of the proposed zero-modified STGNBD along with a comparison of the relative error committed when using the zero-modified STGNBD in place of the STGNBD.