A Saddle Point Approximation for Certain Multivariate Tail Probabilities

We consider the problem of computing the probability of a union of a large number of tail events. This multivariate probability is intractable, so we propose a bound based on univariate and bivariate marginal probabilities, which in turn we approximate by the saddle point method and a correction for skewness. We use this approach to approximate a reliability index for power systems. Our numerical results indicate that this deterministic bound is also an accurate estimate and that it is more efficient than direct Monte Carlo.