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A class of semi-parametric hazard/failure rates with a bathtub shape is ofinterest. It does not only provide a great deal of flexibility over existingparametric methods in the modeling aspect but also results in a closed andtractable Bayes estimator for the bathtub-shaped failure rate (BFR). Such anestimator is derived to be a finite sum over two $\mathbf{S}$-paths due to anexplicit posterior analysis in terms of two (conditionally independent)$\mathbf{S}$-paths. These, newly discovered, explicit results can be proved tobe a Rao-Blackwellization of counterpart results in terms of partitions thatare readily available by a specialization of James (2005)'s work. We developboth iterative and non-iterative computational procedures based on existingefficient Monte Carlo methods for sampling one single $\mathbf{S}$-path.Nmerical simulations are given to demonstrate the practicality and theeffectiveness of our methodology. Last but not least, two applications of theproposed method are discussed, of which one is about a Bayesian test forfailure rates and the other is related to modeling with covariates.

A Bayes method for a Bathtub Failure Rate via two $\mathbf{S}$-paths