Comparison of Numerical Schemes for the Multinormal Integral

The numerical determination of the multinormal integral is required in many problems of statistics and, recently, turned out to be a key problem in structural and operational system reliability [1,2]. For example, the analysis of multi-component systems with multiple failure modes (series — or weakest-link systems) requires the evaluation of the probability of a union of failure events. If equivalent hyperplanes [3] in the so-called standard normal space [4] which represent the modal and/or componental failure sets of the system can be produced, e.g. by FORM/SORM methods, the union probability can be computed as a multinormal integral. As another example the quantification of the redundancy in a system in a probabilistic sense requires the evaluation of intersection probabilities [5,6]. Again, this problem can be reduced to the computation of a multinormal integral if the boundaries of the failure sets can be linearised appropriately. Even if asymptotic second-order approximations of the failure surfaces are used, the evaluation of multinormal integrals is part of the solution to general intersection probabilities [2,7].