The Marshall-Olkin generalized gamma distribution

Abstract

Attempts have been made to define new classes of distributions that provide more flexibility for modelling skewed data in practice. In this work we define a new extension of the generalized gamma distribution (Stacy, The Annals of Mathematical Statistics, 33, 1187-1192, 1962) for Marshall-Olkin generalized gamma (MOGG) distribution, based on the generator pioneered by Marshall and Olkin (Biometrika, 84, 641-652, 1997). This new lifetime model is very flexible including twenty one special models. The main advantage of the new family relies on the fact that practitioners will have a quite flexible distribution to fit real data from several fields, such as engineering, hydrology and survival analysis. Further, we also define a MOGG mixture model, a modification of the MOGG distribution for analyzing lifetime data in presence of cure fraction. This proposed model can be seen as a model of competing causes, where the parameter associated with the Marshall-Olkin distribution controls the activation mechanism of the latent risks (Cooner et al., Statistical Methods in Medical Research, 15, 307-324, 2006). The asymptotic properties of the maximum likelihood estimation approach of the parameters of the model are evaluated by means of simulation studies. The proposed distribution is fitted to two real data sets, one arising from measuring the strength of fibers and the other on melanoma data.