On bayesian estimation of the multiple decrement function in the competing risks problem

Classical methods are inapplicable in estimation problems involving non-identifiable parameters. Bayesian methods, on the other hand, are often both feasible and intuitively reasonable in such problems. This paper establishes the foundations for studying the efficacy of Bayesian updating in estimating nonidentifiable parameters in the competing risks framework. We obtain a useful representation of the posterior distribution of the multiple decrement function, assuming a Dirichler process prior, and derive the limiting posterior distribution. It is noted that posterior estimates of a nonidentifiable parameter may be inferior to estimates based on the prior distribution alone, even when the size of the available sample grows to infinity. This leads, among other things, to the search for distinguished parameter values, or models, in which Bayesian updating necessarily improves upon one's prior estimate. In a companion paper, it is shown that the multivariate exponential distribution can play such a role in the competing risks framework.