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On spherical Monte Carlo simulations for multivariate normal probabilities

Part of: Mathematical finance Probabilistic methods, simulation and stochastic differential equations Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  21 March 2016

Huei-Wen Teng ,
Ming-Hsuan Kang  and
Cheng-Der Fuh
Huei-Wen Teng*
Affiliation:
National Central University
Ming-Hsuan Kang*
Affiliation:
National Chiao Tung University
Cheng-Der Fuh*
Affiliation:
National Central University
*
∗ Postal address: Graduate Institute of Statistics, National Central University, 300 Zhongda Road, Zhongli District, Taoyuan City, 32001, Taiwan, R.O.C.
∗∗∗ Postal address: Department of Applied Mathematics, National Chiao Tung University, 1001 University Road, Hsinchu, 30010, Taiwan, R.O.C.
∗ Postal address: Graduate Institute of Statistics, National Central University, 300 Zhongda Road, Zhongli District, Taoyuan City, 32001, Taiwan, R.O.C.
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Abstract

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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.

Keywords

Sphericalsimulationvariance reductionsphere packingskissing numberlattice

MSC classification

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

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 3 , September 2015 , pp. 817 - 836
Copyright
Copyright © Applied Probability Trust 2015 

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