We consider a coherent system S consisting of m independent components for which we do not know the distributions of the components' lifelengths. If we know the structure function of the system, then we can estimate the distribution of the system lifelength by estimating the distributions of the lifelengths of the individual components. Suppose that we can collect data under the "autopsy model," wherein a system is run until a failure occurs and then the status (functioning or dead) of each component is obtained. This test is repeated n times. The autopsy statistics consist of the age of the system at the time of breakdown and the set of parts that are dead by the time of breakdown. We develop a nonparametric Bayesian estimate of the distributions of the component lifelengths and then use this to obtain an estimate of the distribution of the lifelength of the system. The procedure is applicable to machine-test settings wherein the machines have redundant designs. A parametric procedure is also given.
"Bayesian nonparametric estimation via Gibbs sampling for coherent systems with redundancy." Ann. Statist. 25 (3) 1109 - 1139, June 1997. https://doi.org/10.1214/aos/1069362740