On spherical Monte Carlo simulations for multivariate normal probabilities

TL;DR

This paper introduces a spherical Monte Carlo method for efficiently computing multivariate normal probabilities, combining theoretical variance analysis with practical numerical experiments.

Contribution

It develops a novel spherical Monte Carlo approach with variance bounds and a point set design related to sphere packings, enhancing probability estimation accuracy.

Findings

Variance of the estimator is bounded and minimized using the proposed point set.

The method achieves accurate probability estimates in numerical experiments.

Variance reduction is guaranteed under specific conditions.

Abstract

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