In this paper, we propose an original approach to the computation of the CDF of a continuous function h applied to a real-valued random vector X, when this vector follows an elliptical distribution. Instead of sampling directly from the density of X and computing the proportion of draws for which h(X) exceeds a given threshold c, we propose a level-set approach for calculating the CDF of h(X), by expressing it as a weighted integral computed over the level-sets of its probability density function.

Using Lebesgue integration and the fact that the level-sets of the density function of an elliptical distribution form a family of homothetic hyperellipsoids, we reformulate the probability that h(X) exceeds c as a weighted integral taken over this family of hyperellipsoids, which Lebesgue measure can be computed under closed-form. We further generalize our expression to mixtures of elliptical distributions, and establish potentially fruitful connexions with differential forms by reformulating our integral as a weighted integral over the surface our family of hyperellipsoids using vector fields and the generalized Stokes theorem. Finally, we show that the probability that h(X) exceeds c can be reformulated in terms of the weighted expectation of a random function of U, where U  follows a uniform distribution on the unit sphere, and provide a consistent estimator for this probability.

To assess the performance of our method, we compare it to a reference method, consisting in directly drawing a sample from the density function of  and computing the frequency of the event (h(X) > c). We carry out comparisons for univariate polynomials of random variables, where the random vector X has dimension equal to 3, 6 and 10. We assume X follows a multivariate Gaussian distribution, which allows us to directly express the probability density of h(X) under closed-form and compute the exact probability of (h(X) > c). This allows us to compute the empirical bias and variance associated to both approaches. Our results suggest our method consistently exhibits lower empirical bias and variance compared to the reference method, i.e. it converges faster to the true value.