Zero Failure Data

Estimating failure probabilities or risk when no failures have been observed is an inherently difficult, but not uncommon task. Several approaches are described in this article. Be content with a “pessimistic” estimate—an upper bound or a conservative distribution. This can be a confidence limit, or the results from a Bayesian distribution whose percentiles match the confidence upper limits, or the results based on a “noninformative prior”. Find data for similar equipment, and construct an estimate with the aid of the other data. Empirical Bayes or hierarchical Bayes methods may be appropriate. (This method is not limited to zero failure data.) If one must fall back on engineering judgment, a “constrained noninformative prior” may be helpful. Analyze the parts, ways, or processes by which the system can fail, and use data on these various parts to construct an estimate for the system as a whole. Several examples are mentioned, including structural reliability and various segments of probabilistic safety assessment (PSA). If a few failures have been observed instead of no failures, some aspects of the situation are greatly improved: we no longer need to be content with an upper bound, and Bayesian results are much less sensitive to details of the prior distribution. Keywords: confidence upper bounds; Bayesian upper limits; Jeffreys noninformative prior; empirical Bayes; hierarchical Bayes; probabilistic safety assessment (PSA); probabilistic risk assessment (PRA)