Sang Gil Kang

Noninformative Priors for the Ratio of the Scale Parameters in the Inverted Exponential Distributions

Abstract In this paper, we develop the noninformative priors for the ratio of the scale parameters in the invertedexponential distributions. The first and second order matching priors, the reference prior and Jeffreys prior aredeveloped. It turns out that the second order matching prior matches the alternative coverage probabilities, is acumulative distribution function matching prior and is a highest posterior density matching prior. In addition, thereferencepriorandJeffreys’priorarethesecondordermatchingprior. Weshowthattheproposedreferencepriormatches the target coverage probabilities in a frequentist sense through a simulation study as well as provide anexample based on real data is given.Keywords: Inverted exponential distribution, matching prior, reference prior, scale parameter. 1. Introduction Exponential distribution is the most exploited distribution for lifetime data analysis. However, itssuitabilityisrestrictedtoaconstanthazardrate,whichisdifficulttojustifyinmanypracticalproblems.This leads to the development of alternative models for lifetime data. A number of distributions suchasWeibullandgammahavebeenextensivelyusedtoanalyzelifetimedatasuchassituationswherethehazard rate is monotonically increasing or decreasing; however, the non-monotonicity of the hazardrate has also been observed in many situations. For example, the hazard rate initially increases withtime and reaches a peak after some finite period of time and then declines slowly in some studies ofmortality associated with particular diseases. Thus, the need to analyze such data whose hazard rateis non-monotonic was realized and suitable models were proposed. Killer and Kamath (1982), Lin

Noninformative Priors for the Stress-Strength Reliability in the Generalized Exponential Distributions

Abstract This paper develops the noninformative priors for the stress-strength reliability from one parameter gener-alized exponential distributions. When this reliability is a parameter of interest, we develop the first, secondorder matching priors, reference priors in its order of importance in parameters and Jeffreys’ prior. We revealthat these probability matching priors are not the alternative coverage probability matching prior or a highestposterior density matching prior, a cumulative distribution function matching prior. In addition, we reveal thatthe one-at-a-time reference prior and Jeffreys’ prior are actually a second order matching prior. We show thatthe proposed reference prior matches the target coverage probabilities in a frequentist sense through a simulationstudy and a provided example.Keywords: Generalized exponential model, matching prior, reference prior, stress-strength relia-bility. 1. Introduction The one parameter generalized exponential distribution was introduced by Gupta and Kundu (1999)as an alternative to the gamma or Weibull distributions for analyzing lifetime data (Gupta and Kundu,2001). An advantage of employing the generalized exponential distribution is that the distributionfunction can be obtained in a closed form. Kundu and Gupta (2007) showed that the generalizedexponentialdistributionisquiteflexibleandcanbeusedveryeffectivelyinanalyzingpositivelifetimedata in place of the gamma or Weibull models. Raqab and Madi (2005) studied the Bayeian inferencefor the parameters and reliability function.Consider