Bayesian Analysis of the Mixing Function in a Mixture of Two Exponential Distributions

SUMMARY. In this paper we propose a semiparamteric Bayesian approach to estimate the mixing function in a mixture of two exponential distributions. Unlike, the traditional mixture of two distributions in this paper we assume that the mixing parameter changes with time. Such models arise naturally in many applications such as software reliability engineering and other related elds. Our proposed models are diierent from the usual mixture of two or more probability distributions in the sense that we consider mixing two or more hazard rates instead of probability densities. This makes the underlying parameters of the model more interpretable from an engineering application point of view as most engineers think in terms of hazard rates instead of probability densities. The Bayesian posterior distributions of the parameters are analytically intractable. We use Markov Chain Monte Carlo (MCMC) methods to t our proposed models to observed data. Another advantage of the simulation based Bayesian methods for our models is that censoring can be handled routinely. We also use a decision-theoretic model choice criteria to compare our proposed semiparametric model with a corresponding fully parametric model. We illustrate our approach with an application involving insulating uid failure data.