Diverse Applications of Two Weibull ExtensionsGovind

In this talk we review several applications of two Weibull extensions in statistical theory and applied research. The two extensions are the generalized Weibull and the exponentiated Weibull families, and the applications are in the areas of reliability and survival analysis, exteme value analysis, isotones and distributional approximations. 1 The Generalized Weibull Family The generalized Weibull family, rst suggested in Mudholkar et al. (1991) for constructing isotones, has the following quantile function Q(u), and the distribution function F (x): Q(u) = (1 ? (1 ? u))==] 6 = 0; ? log(1 ? u)] = 0; (1) and F (x) = 1 ? (1 ? (x==) 1==) 1== ; (2) where ; > 0 and ?1 < < 1. The generalized Weibull family yields the Weibull family when = 0, the exponential distribution for = 1, = 0, and the log-logistic distribution for = ?1, which is often used as a model in survival studies. Moreover, common parametric distributions such as the lognormal and the gamma distributions, are very well approximated by members of the family; see Mudholkar and Kollia (1994). Further analysis of the generalized Weibull family, including examination of the skewness and kurtosis, density shapes and tail characteristics, extreme value distributions, density classiication, and its relation to the Pearson system and other distributions can be found in Mudholkar and Kollia (1994). 2 The Exponentiated Weibull Family Originally proposed by Mudholkar and Srivastava (1993) in the context of bathtub shaped failure rate data, the exponentiated Weibull family has quantile function, Q(u) = ? log(1 ? u 1==)] 1== ; (3) and distribution function, F (x) = 1 ? exp(?(x==))] ; (4) where ; ; > 0. Distributional properties, including skewness and kurtosis, density shapes and tail character, and the extreme value and extreme spacings distributions for the members of the exponentiated Weibull family are in Mudholkar and Hutson (1996), which also includes an application to analysis of extremes using ood data. More recent references include Gupta et al.