A comparison of the generalized gamma and exponentiated Weibull distributions

This paper provides a comparison of the three-parameter exponentiated Weibull (EW) and generalized gamma (GG) distributions. The connection between these two different families is that the hazard functions of both have the four standard shapes (increasing, decreasing, bathtub, and arc shaped), and in fact, the shape of the hazard is the same for identical values of the three parameters. For a given EW distribution, we define a matching GG using simulation and also by matching the 5 ^th, 50 ^th, and 95 ^th percentiles. We compare EW and matching GG distributions graphically and using the Kullback-Leibler distance. We find that the survival functions for the EW and matching GG are graphically indistinguishable, and only the hazard functions can sometimes be seen to be slightly different. The Kullback-Leibler distances are very small and decrease with increasing sample size. We conclude that the similarity between the two distributions is striking, and therefore, the EW represents a convenient alternative to the GG with the identical richness of hazard behavior. More importantly, these results suggest that having the four basic hazard shapes may to some extent be an important structural characteristic of any family of distributions.