## A novel series expansion for the multivariate normal probability integrals based on Fourier series

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- by Hatem A. Fayed and Amir F. Atiya;
- Math. Comp.
**83**(2014), 2385-2402 - DOI: https://doi.org/10.1090/S0025-5718-2014-02844-5
- Published electronically: April 17, 2014
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## Abstract:

In this article, we derive a series expansion of the multivariate normal probability integrals based on Fourier series. The basic idea is to transform the limits of each integral from $h_i$ to $\infty$ to be from $-\infty$ to $\infty$ by multiplying the integrand by a periodic square wave that approximates the domain of the integral. This square wave is expressed by its Fourier series expansion. Then a Cholesky decomposition of the covariance matrix is applied to transform the integrand to a simple one that can be easily evaluated. The resultant formula has a simple pattern that is expressed as multiple series expansion of trigonometric and exponential functions.## References

- M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1964.
- I. Deák,
*Three digit accurate multiple normal probabilities*, Numer. Math.**35**(1980), no. 4, 369–380. MR**593834**, DOI 10.1007/BF01399006 - István Deák,
*Random number generators and simulation*, Mathematical Methods of Operations Research, vol. 4, Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest, 1990. Translated and revised from the Hungarian by the author. MR**1080965** - D. R. Divgi,
*Calculation of univariate and bivariate normal probability functions*, Ann. Statist.**7**(1979), no. 4, 903–910. MR**532253** - T.G. Donnelly, Algorithm 462: Bivariate normal distribution, Commun. ACM 16 (1973) 636.
- Z. Drezner,
*Computation of the bivariate normal integral*, Math. Comp.**32**(1978), no. 141, 277–279. MR**461849**, DOI 10.1090/S0025-5718-1978-0461849-9 - Zvi Drezner and G. O. Wesolowsky,
*On the computation of the bivariate normal integral*, J. Statist. Comput. Simulation**35**(1990), no. 1-2, 101–107. MR**1041725**, DOI 10.1080/00949659008811236 - Hatem A. Fayed and Amir F. Atiya,
*An evaluation of the integral of the product of the error function and the normal probability density with application to the bivariate normal integral*, Math. Comp.**83**(2014), no. 285, 235–250. MR**3120588**, DOI 10.1090/S0025-5718-2013-02720-2 - J. Gai, A computational study of the bivariate normal probability function, M.Sc. thesis, Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada, 2002.
- H. I. Gassmann,
*Multivariate normal probabilities: implementing an old idea of Plackett’s*, J. Comput. Graph. Statist.**12**(2003), no. 3, 731–752. MR**2005459**, DOI 10.1198/1061860032283 - A. Genz, Numerical computation of multivariate normal probabilities, J. Comp. Graph. Stat. 1 (1992) 141-150.
- A. Genz, Comparison of methods for the computation of multivariate normal probabilities, Comp. Sci. Stat. 25 (1993) 400-405.
- Alan Genz,
*Numerical computation of rectangular bivariate and trivariate normal and $t$ probabilities*, Stat. Comput.**14**(2004), no. 3, 251–260. MR**2086401**, DOI 10.1023/B:STCO.0000035304.20635.31 - Alan Genz and Frank Bretz,
*Computation of multivariate normal and $t$ probabilities*, Lecture Notes in Statistics, vol. 195, Springer, Dordrecht, 2009. MR**2840595**, DOI 10.1007/978-3-642-01689-9 - Shanti S. Gupta,
*Probability integrals of multivariate normal and multivariate $t$*, Ann. Math. Statist.**34**(1963), 792–828. MR**152068**, DOI 10.1214/aoms/1177704004 - Bernard Harris and Andrew P. Soms,
*The use of the tetrachoric series for evaluating multivariate normal probabilities*, J. Multivariate Anal.**10**(1980), no. 2, 252–267. MR**575928**, DOI 10.1016/0047-259X(80)90017-2 - N.L. Johnson, S. Kotz, Distributions in statistics: Continuous multivariate distribution, John Wiley and Sons, New York, 1972.
- M. G. Kendall,
*Proof of relations connected with the tetrachoric series and its generalization*, Biometrika**32**(1941), 196–198. MR**5573**, DOI 10.1093/biomet/32.2.196 - Tetsuhisa Miwa, A. J. Hayter, and Satoshi Kuriki,
*The evaluation of general non-centred orthant probabilities*, J. R. Stat. Soc. Ser. B Stat. Methodol.**65**(2003), no. 1, 223–234. MR**1959823**, DOI 10.1111/1467-9868.00382 - Donald B. Owen,
*Tables for computing bivariate normal probabilities*, Ann. Math. Statist.**27**(1956), 1075–1090. MR**127562**, DOI 10.1214/aoms/1177728074 - K. Pearson, Mathematical contributions to the theory of evolution. VII. on the correlation of characters not quantitatively. Philos. Trans. R. Soc. S-A. 196 (1901) 1-47.
- A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev,
*Integrals and series. Vol. 1*, Gordon & Breach Science Publishers, New York, 1986. Elementary functions; Translated from the Russian and with a preface by N. M. Queen. MR**874986** - N. G. Shephard,
*From characteristic function to distribution function: a simple framework for the theory*, Econometric Theory**7**(1991), no. 4, 519–529. MR**1151944**, DOI 10.1017/S0266466600004746 - Tamás Szántai,
*An algorithm for determining the values of multivariate normal distributions and their gradients*, Alkalmaz. Mat. Lapok**2**(1976), no. 1-2, 27–39 (Hungarian, with English summary). MR**438537** - Y. L. Tong,
*The multivariate normal distribution*, Springer Series in Statistics, Springer-Verlag, New York, 1990. MR**1029032**, DOI 10.1007/978-1-4613-9655-0

## Bibliographic Information

**Hatem A. Fayed**- Affiliation: Department of Engineering, Mathematics and Physics, Faculty of Engineering, Cairo University, Cairo, Egypt 12613
- Email: h{_}fayed@eng.cu.edu.eg
**Amir F. Atiya**- Affiliation: Department of Computer Engineering, Faculty of Engineering, Cairo University, Cairo, Egypt 12613
- Email: amir@alumni.caltech.edu
- Received by editor(s): June 28, 2012
- Received by editor(s) in revised form: January 9, 2013
- Published electronically: April 17, 2014
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**83**(2014), 2385-2402 - MSC (2010): Primary 42A16, 62H86
- DOI: https://doi.org/10.1090/S0025-5718-2014-02844-5
- MathSciNet review: 3223336