Hazard Rates and Generalized Beta Distributions

Abstract

This paper considers the behavior of the hazard rates of the Generalized gamma, and beta of the first and second kind distributions. The hazard functions include strictly decreasing, constant, strictly increasing, ∼ or ⋂ shaped hazard rates. By considering the generalized distributions a unified development for such distributions as beta type 1, beta type 2, Burr types 3 and 12, power, Weibull, gamma, Lomax, Fisk, uniform, Rayleigh, and exponential are included as special cases. The results are conveniently summarized in three figures.

The generalized distributions considered in this paper are seen to provide models for all of the different shaped hazard rates mentioned above. This flexibility permits the data to determine the nature of the hazard function without its being inadvertently imposed through the selection of an improper model. For example, the selection of a Weibull distribution permits a decreasing, constant, or increasing hazard rate, but not a ⋃ or ⋂ shaped one. The use of the generalized gamma or either of the generalized beta functions considered in section II does permit realization of these additional shapes for the hazard rate.