THE EXPONENTIATED WEIBULL FAMILY - A REANALYSIS OF THE BUS-MOTOR-FAILURE DATA

The Weibull family with survival function exp{-(y/sigma)(alpha)}, for alpha > 0 and y greater than or equal to 0, is generalized by introducing an additional shape parameter theta. The space of shape parameters alpha > 0 and theta > 0 can be divided by boundary line alpha = 1 and curve alpha theta = 1 into four regions over which the hazard function is, respectively, increasing, bathtub-shaped, decreasing, and unimodal. The new family is suitable for modeling data that indicate nonmonotone hazard rates and can be adopted for testing goodness of fit of Weibull as a submodel. The usefulness and flexibility of the family is illustrated by reanalyzing five classical data sets on bus-motor failures from Davis that are typical of data in repair-reuse situations and Efron's data pertaining to a head-and-neck-cancer clinical trial. These illustrative data involve censoring and indicate bathtub, unimodal, and increasing but possibly non-Weibull hazard-shape models.