Inference for the First Order Statistic Distribution Based on Masked Exponential Distributions for Two Independent Competing Risks

This paper considers the scenario of independent risks which have times to occurrence described by Exponential distributions. The problem involves masking, that is, the time of occurrence of the earlier risk (first order statistic) is observed but not the specific risk. Nonidentifiability is a consequence of this limitation, and a Bayesian procedure is implemented to address this issue. More specifically, independent Gamma priors are placed on the Exponential parameters which allow identification of the Exponential parameters. Confidence intervals (CIs) for the distribution function F((1) ) (x) of the time to first occurrence are studied. Rigorous computer simulations are carried out to assess the relative performance of CIs based on Maximum Likelihood methods, Bayesian, and distribution-free asymptotics in terms of empirical coverage probability and average interval length. All the approaches presented in this study are for two independent Exponential risks, which can be generalized to three or more risks