On spherical Monte Carlo simulations for multivariate normal probabilities

摘要

The calculation of multivariate normal probabilities is of great importancein many statistical and economic applications. This paper proposes a sphericalMonte Carlo method with both theoretical analysis and numerical simulation.First, the multivariate normal probability is rewritten via an inner radialintegral and an outer spherical integral by the spherical transformation. Forthe outer spherical integral, we apply an integration rule by randomly rotatinga predetermined set of well-located points. To find the desired set, we derivean upper bound for the variance of the Monte Carlo estimator and propose a setwhich is related to the kissing number problem in sphere packings. For theinner radial integral, we employ the idea of antithetic variates and identifycertain conditions so that variance reduction is guaranteed. Extensive MonteCarlo experiments on some probabilities calculation confirm these claims.