Advances in Applied Probability

On spherical Monte Carlo simulations for multivariate normal probabilities

Huei-Wen Teng, Ming-Hsuan Kang, and Cheng-Der Fuh

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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

Permanent link to this document
https://projecteuclid.org/euclid.aap/1444308883

Digital Object Identifier
doi:10.1239/aap/1444308883

Mathematical Reviews number (MathSciNet)
MR3406609

Zentralblatt MATH identifier
06505269

Subjects
Primary: 11K45: Pseudo-random numbers; Monte Carlo methods 65C05: Monte Carlo methods
Secondary: 65C60: Computational problems in statistics 91G60: Numerical methods (including Monte Carlo methods) 91G70: Statistical methods, econometrics

Keywords
Spherical simulation variance reduction sphere packings kissing number lattice

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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