Bayes estimation of P(Y < X) for the Weibull distribution with arbitrary parameters

Abstract

• We focus on Bayes estimation of R = P ( Y < X ) when X and Y are different Weibull random variables. • The closed form of R is derived. • The Bayes estimate and credible interval for R are presented. • The analysis of two real data sets show the significance and necessity of our study. In the model of R = P ( Y < X ) , X and Y usually represent the strength of a system and stress applied to it. Then, R is the measure of system reliability. In this paper, Bayes estimation of R = P ( Y < X ) is studied under the assumption that X and Y are independent Weibull random variables with arbitrary scale and shape parameters. We show here for the first time how to compute the Bayes estimates and credible intervals for R in that case. First, a closed form expression for R is derived. Prior distributions are assumed for Weibull parameters, and the posterior distribution is presented. Next, by proposing an universal sample-based method according to the Monte Carlo Markov Chain (MCMC) method, we draw samples and compute the Bayes estimates and credible intervals for R . Through Monte Carlo simulations and two real data examples, the proposed method is demonstrated to be robust and satisfactory.