### On spherical Monte Carlo simulations for multivariate normal probabilities

#### Abstract

The calculation of multivariate normal probabilities is of great importance in many statistical and economic applications. In this paper we propose a spherical Monte Carlo method with both theoretical analysis and numerical simulation. We start by writing the multivariate normal probability via an inner radial integral and an outer spherical integral using 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 simulations on some probabilities confirm these claims.

#### Article information

Source
Adv. in Appl. Probab., Volume 47, Number 3 (2015), 817-836.

Dates
First available in Project Euclid: 8 October 2015

https://projecteuclid.org/euclid.aap/1444308883

Digital Object Identifier
doi:10.1239/aap/1444308883

Mathematical Reviews number (MathSciNet)
MR3406609

Zentralblatt MATH identifier
06505269

#### Citation

Teng, Huei-Wen; Kang, Ming-Hsuan; Fuh, Cheng-Der. On spherical Monte Carlo simulations for multivariate normal probabilities. Adv. in Appl. Probab. 47 (2015), no. 3, 817--836. doi:10.1239/aap/1444308883. https://projecteuclid.org/euclid.aap/1444308883

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