September 2015 On spherical Monte Carlo simulations for multivariate normal probabilities
Huei-Wen Teng, Ming-Hsuan Kang, Cheng-Der Fuh
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Adv. in Appl. Probab. 47(3): 817-836 (September 2015). DOI: 10.1239/aap/1444308883


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.


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Huei-Wen Teng. Ming-Hsuan Kang. Cheng-Der Fuh. "On spherical Monte Carlo simulations for multivariate normal probabilities." Adv. in Appl. Probab. 47 (3) 817 - 836, September 2015.


Published: September 2015
First available in Project Euclid: 8 October 2015

zbMATH: 06505269
MathSciNet: MR3406609
Digital Object Identifier: 10.1239/aap/1444308883

Primary: 11K45 , 65C05
Secondary: 65C60 , 91G60 , 91G70

Keywords: kissing number , lattice , simulation , sphere packings , Spherical , variance reduction

Rights: Copyright © 2015 Applied Probability Trust


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Vol.47 • No. 3 • September 2015
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