On probabilistic safety assessment in the case of zero failures

Probabilistic safety assessment often depends on the quantification of the frequencies of failures, on the basis of zero failures in the available data. A variety of estimates for such frequencies have been suggested in the literature, mainly within the classical and Bayesian statistical frameworks, and with emphasis on the constant failure rate of an exponential distribution. In this paper, this problem is considered from the perspective of non-parametric predictive inference (NPI), where the main difference from the established approaches is the use of lower and upper probabilities, leading to lower and upper estimates for the constant exponential failure rate instead of point estimates. On the basis of zero failures observed, the lower estimate for the failure rate is zero; so interest is mostly in the upper failure rate. Such lower and upper estimates reflect the indeterminacy due to the data scarcity. The NPI-based upper estimate for the failure rate depends not only on the length of the failure-free period represented as data but also on the required or chosen length of the future period considered in the risk assessment, in a possibly surprising manner. The use of failure rate estimates in probabilistic safety assessment is discussed, and a more straightforward predictive approach is suggested.