THE GAMMA-WEIBULL DISTRIBUTION

An extension of the Weibull distribution which involves an additional shape parameter is being proposed. Interestingly, the additional parameter acts somewhat as a location parameter while the support of the distribution remains the positive half-line. Since the gamma distribution is a particular case of this distribution, the latter is referred to as a gamma-Weibull distribution. The gamma-Weibull distribution is in fact a reparameterization of the generalized gamma distribution, which has received little attention in recent years. Some parameters of the gamma-Weibull model have a more straightforward interpretation than those associated with the generalized gamma distribution. Moreover, the gamma-Weibull distribution does not contain a threshold parameter. Accordingly, it readily lends itself to various estimation methodologies and exhibits regular asymptotics. Numerous distributions such as the Rayleigh, half-normal and Maxwell distributions can also be obtained as special cases. The moment generating function of a gamma -Weibull random variable is derived by making use of the inverse Mellin transform technique and expressed in terms of generalized hypergeometric functions. This provides computable representations of the moment generating functions of several of the distributions that were identified as particular cases. Other statistical functions such as the cumulative distribution function of a gamma-Weibull random variable, its moments, hazard rate and associated entropy are also given in closed form. The proposed reparametrization is utilized to model two data sets. The gamma-Weibull distribution provides a better fit than the two parameter Weibull model or its shifted counterpart, as measured by the Anderson-Darling and Cramer-von Mises statistics.