Closeness of Gamma and Generalized Exponential Distribution

Abstract

Recently a new distribution, named as generalized exponential distribution or exponentiated exponential distribution was introduced and studied quite extensively by the authors. It is observed that the generalized exponential distribution can be used as an alternative to the gamma distribution in many situations. Different properties like monotonicity of the hazard functions and tail behaviors of the gamma distribution and the generalized exponential distribution are quite similar in nature, but the later one has a nice compact distribution function. It is observed that for a given gamma distribution there exists a generalized exponential distribution so that the two distribution functions are almost identical. Since the gamma distribution function does not have a compact form, efficiently generating gamma random numbers is known to be problematic. We observe that for all practical purposes it is possible to generate approximate gamma random numbers using generalized exponential distribution and the random samples thus obtained cannot be differentiated using any statistical tests. Moreover, if there is a skewed data set where gamma distribution fits very well, the generalized exponential distribution also can be used. We use two real life data sets and observe that the fitted distribution functions are “almost identical” in many respects in both the cases.

Keywords:

• Distribution function
• Hazard function
• Kolmogorov-Smirnov test
• Kullbuck–Leibler discrepancy measure
• Polygamma function

Acknowledgments

The authors would like to thank the referee for some helpful suggestions. Part of the work of the first author was supported by a grant from the Natural Sciences and Engineering Research Council.