The Exponentiated Weibull Family: A Reanalysis of the Bus-Motor-Failure Data

Abstract

The Weibull family with survival function exp{-(y/σ)α}, for α > 0 and y ≥ 0, is generalized by introducing an additional shape parameter θ. The space of shape parameters α > 0 and θ > 0 can be divided by boundary line α = 1 and curve (αθ = 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 datapertainingtoahead-and-neck-cancerclinical trial. These illustrative datainvolvecensoring and indicate bathtub, unimodal, and increasing but possibly non-Weibull hazard-shape models.