Bayes Estimation of Reliability Using an Estimated Prior Distribution

Abstract

Suppose that the conditional failure time distribution F ( t ∣ θ) depends on a random parameter θ whose probability distribution G (θ) is unknown. The unconditional failure time distribution is F G ( f ) = ∫ F ( t ∣ θ) dG (θ). In this paper we consider the estimation of G in reliability models when a priori information about the parameter θ is specified in the form of an initial guess, G 0 , of G . Utilizing the concepts of Dirichlet process priors on G , a Bayes estimate F̂ G of F G may be obtained based on k observed lifetimes from F G . Then an estimate Ĝ k of G is found from F̂ G using a linear programming approach. For the Weibull failure time distribution F ( t ∣ θ) with random scale parameter θ, the effect of using the estimated prior Ĝ k in Bayes estimation of reliability is studied by Monte Carlo simulations.