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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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

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

First available in Project Euclid: 8 October 2015

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Zentralblatt MATH identifier

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

Spherical simulation variance reduction sphere packings kissing number lattice


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.

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  • Anderson, T. W., Olkin, I. and Underhill, L. G. (1987). Generation of random orthogonal matrices. SIAM J. Sci. Statist. Comput. 8, 625–629.
  • Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.
  • Conway, J. H. and Sloane, N. J. A. (1999). Sphere Packings, Lattices and Groups, 3rd edn. Springer, New York.
  • Craig, P. (2008). A new reconstruction of multivariate normal orthant probabilities. J. R. Statist. Soc. B 70, 227–243.
  • Davis, P. J. and Rabinowitz, P. (1984). Methods of Numerical Integration, 2nd edn. Academic Press, Orlando, FL.
  • Deák, I. (1980). Three digit accurate multiple normal probabilities. Numer. Math. 35, 369–380.
  • Deák, I. (2000). Subroutines for computing normal probabilities of sets–-computer experiences. Ann. Operat. Res. 100, 103–122.
  • Diaconis, P. and Shahshahani, M. (1987). The subgroup algorithm for generating uniform random variables. Prob. Eng. Inf. Sci. 1, 15–32.
  • Fang, K.-T. and Wang, Y. (1994). Number-Theoretic Methods in Statistics (Monogr. Statist. Appl. Prob. 51). Chapman & Hall, London.
  • Genz, A. (1992). Numerical computation of multivariate normal probabilities. J. Comput. Graphical Statist. 1, 141–150.
  • Genz, A. (1993). Comparison of methods for the computation of multivariate normal probabilities. Comput. Sci. Statist. 25, 400–405.
  • Genz, A. and Bretz, F. (2002). Methods for the computation of multivariate $t$ probabilities. J. Comput. Graphical Statist. 11, 950–971.
  • Genz, A. and Bretz, F. (2009). Computation of Multivariate Normal and $t$ Probabilities (Lecture Notes Statist. 195). Springer, Dordrecht.
  • Glasserman, P., Heidelberger, P. and Shahabuddin, P. (2000). Variance reduction techniques for estimating value-at-risk. Manag. Sci. 46, 1349–1364.
  • Gupton, G. M., Finger, C. C. and Bhatia, M. (1997). CreditMetrics – Technical Document. Morgan, New York.
  • Hajivassiliou, V., McFadden, D. and Ruud, P. (1996). Simulation of multivariate normal rectangle probabilities and their derivatives: theoretical and computational results. J. Econometrics 72, 85–134.
  • Heiberger, R. M. (1978). Algorithm AS 127: generation of random orthogonal matrices. J. R. Statist. Soc. C 27, 199–206.
  • Hsu, J. C. (1996). Multiple Comparisons. Chapman & Hall, London.
  • Miwa, T., Hayter, A. J. and Kuriki, S. (2003). The evaluation of general non-centred orthant probabilities. J. R. Statist. Soc. B 65, 223–234.
  • Monahan, J. and Genz, A. (1997). Spherical-radial integration rules for Bayesian computation. J. Amer. Statist. Assoc. 92, 664–674.
  • Nebe, G. and Sloane, N. J. A. A catalogue of lattices. Available at$\sim$Gabriele.Nebe/ LATTICES/.
  • Ross, S. M. (2013). Simulation, 5th edn. Elsevier, Amsterdam.
  • Sándor, Z. and András, P. (2004). Alternative sampling methods for estimating multivariate normal probabilities. J. Econometrics 120, 207–234.
  • Sloane, N. J. A. Spherical Codes. Available at$\#$I.
  • Somerville, P. N. (2001). Numerical computation of multivariate normal and multivariate $t$ probabilities over ellipsoidal regions. J. Statist. Software 6, 10pp.
  • Stewart, G. W. (1980). The efficient generation of random orthogonal matrices with an application to condition estimators. SIAM J. Numerical Anal. 17, 403–409.
  • Vijverberg, W. P. M. (1997). Monte Carlo evaluation of multivariate normal probabilities. J. Econometrics 76, 281–307. \endharvreferences