Multiple tail median approach for high reliability estimation

Sampling-based reliability estimation with expensive computer models may be computationally prohibitive when the probability of failure is low (or high reliability). One way to alleviate the computational expense is to extrapolate reliability estimates from observed levels to unobserved levels. Classical tail modeling techniques, two of which are discussed in this paper provide extrapolation models using asymptotic theory by approximating the tail of the cumulative distribution function (CDF). This paper proposes three additional tail extrapolation techniques in performance space. The proposed tail extrapolations are based on the application of nonlinear transformation to the CDF of the performance measure. The proposed approach called the multiple tail median employs all the five techniques simultaneously and uses the median as the best estimate. The range of the five estimates is used as an estimate of the order of magnitude of error in the median. The method is demonstrated on four standard statistical distributions and two engineering examples. It is found that the best tail model changes for different distributions. Also, for the same distribution no single model performed best at different extrapolation levels. Thus, no single tail model is preferable. We also find that the median is usually much closer to the best of the five estimates than to the worst, and that the range mostly varies between 2 and 10 times the magnitude of the error in the median. Therefore the median estimate serves as insurance against bad predictions if one was to use a single estimate. For the examples studied, the use of tail modeling reduced the number of samples required for given accuracy by one to three orders of magnitude.