Hierarchical Bayes estimation for the exponential-multinomial model in reliability and competing risks

The exponential-multinomial distribution arises from: (1) observing the system failure of a series system with p components having independent exponential lifetimes, or (2) a competing-risks model with p sources of failure, as well as (3) the Marshall-Olkin multivariate exponential distribution under a series sampling scheme. Hierarchical Bayes (HB) estimators of the component sub-survival function and the system reliability are obtained using the Gibbs sampler. A large-sample approximation of the posterior pdf is used to derive the HB estimators of the parameters of the model with respect to the quadratic loss function. The exact risk of the HE estimator is obtained and is compared with those corresponding to some other estimators such as Bayes, maximum likelihood, and minimum variance unbiased estimators.