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

TL;DR

This paper introduces a Bayesian semi-parametric approach for estimating bathtub-shaped failure rates using explicit posterior analysis with two S-paths, offering flexible modeling and efficient computation.

Contribution

It presents a novel Bayesian estimator for bathtub failure rates based on two S-paths, with explicit formulas and practical Monte Carlo algorithms.

Findings

The estimator is a finite sum over two S-paths.

Numerical simulations confirm the method's effectiveness.

Applications include Bayesian failure rate testing and covariate modeling.

Abstract

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