Estimating the cumulative downtime distribution of a highly reliable component

The distribution of the cumulative downtime for a highly reliable component over an interval of time is approximated using a compound Bernoulli process. Given a set of observed cumulative downtimes, the maximum likelihood (ML) and uniformly minimum variance unbiased (UMVU) estimator of the approximate cumulative downtime distribution are derived under the assumption of exponential repair times and are compared to the nonparametric estimator. The ML estimator is more efficient than the UMVU estimator which itself is more efficient than the nonparametric estimator.