Generalization of the Weibull distribution: the odd Weibull family

A three-parameter generalization of the Weibull distribution is presented to deal with general situations in modeling survival process with various shapes in the hazard function.This generalized Weibull distribution will be referred to as the odd Weibull family, as it is derived by considering the distributions of the odds of the Weibull and inverse Weibull families. As a result, the odd Weibull family is not only useful for testing goodness-of-fit of the Weibull and inverse Weibull as submodels, but it is also convenient for modeling and fitting different data sets, especially in the presence of censoring. The model parameters for uncensored data are estimated in two different ways because of the fact that the inverse transformation of the odd Weibull family does not change its density function. Adequacy of the model for the given uncensored data is illustrated by using the plot of scaled fitted total time on test (TTT) transforms. Furthermore, simulation studies are conducted to measure the discrepancy between empirical and fitted TTT transforms by using a previously proposed test statistic. Three different examples are, respectively, provided based on data from survival, reliability and environmental sciences to illustrate increasing, bathtub and unimodal failure rates.