Bayes Estimation of Reliability Using an Estimated Prior Distribution

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, G₀, 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.