Default Bayesian testing for the equality of shape parameters in the inverse Weibull distributions

Abstract

Abstract This article deals with the problem of testing for the equality of the shape pa-rameters in two inverse Weibull distributions. We propose Bayesian hypothesis testingprocedures for the equality of the shape parameters under the noninformative prior.The noninformative prior is usually improper which yields a calibration problem thatmakes the Bayes factor to be de ned up to a multiplicative constant. So we propose thedefault Bayesian hypothesis testing procedures based on the fractional Bayes factor andthe intrinsic Bayes factors under the reference priors. Simulation study and an exampleare provided.Keywords: Fractional Bayes factor, intrinsic Bayes factor, inverse Weibull distribution,reference prior, shape parameter. 1. Introduction Consider Xhas an inverse Weibull distribution with the scale parameter and the shapeparameter . Then the likelihood function isL(; ) =  x ( +1) expˆx˙;x>0; (1.1)where >0 and >0. Drapella (1993) calls the inverse Weibull distribution as the comple-mentary Weibull distribution and Mudhokar and Kollia (1994) call it the reciprocal Weibulldistribution. The inverse Weibull distribution has the ability to model failure rates whichare quite common in reliability (Keller and Kamath, 1982; Erto, 1989; Calabria and Pulcini,1989). The density and the hazard function of inverse Weibull can be unimodal or decreas-ing depending on the choice of the shape parameter. Also the inverse Weibull distributionbecomes the inverse Rayleigh distribution and the inverse exponential distribution for = 2and = 1, respectively.The present paper focuses on Bayesian testing for the equality of the shape parametersin two inverse Weibull distributions. In Bayesian model selection or testing problem, theBayes factor under proper priors or informative priors have been very successful. However,