An evaluation of the integral of the product of the error function and the normal probability density with application to the bivariate normal integral

Authors:
Hatem A. Fayed and Amir F. Atiya

Journal:
Math. Comp. **83** (2014), 235-250

MSC (2010):
Primary 33B20, 33C45

DOI:
https://doi.org/10.1090/S0025-5718-2013-02720-2

Published electronically:
May 29, 2013

MathSciNet review:
3120588

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper derives the value of the integral of the product of the error function and the normal probability density as a series of the Hermite polynomial and the normalized incomplete Gamma function. This expression is beneficial, and can be used for evaluating the bivariate normal integral as a series expansion. This expansion is a good alternative to the well-known tetrachoric series, when the correlation coefficient, , is large in absolute value.

**1.**M. Abramowitz, I.A. Stegun, Handbook of mathematical functions, Dover, New York, 1964.**2.**Larry C. Andrews,*Special functions for engineers and applied mathematicians*, Macmillan Co., New York, 1985. MR**779819****3.**Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.),*NIST handbook of mathematical functions*, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR**2723248****4.**K. Briggs, http://keithbriggs.info/documents/erf-integrals.pdf, 2003.**5.**Jan W. Dash,*Quantitative finance and risk management*, World Scientific Publishing Co., Inc., River Edge, NJ, 2004. A physicist’s approach. MR**2090677****6.**D. R. Divgi,*Calculation of univariate and bivariate normal probability functions*, Ann. Statist.**7**(1979), no. 4, 903–910. MR**532253****7.**T.G. Donnelly, Algorithm 462: Bivariate normal distribution, Commun. ACM 16 (1973), 636.**8.**Z. Drezner,*Computation of the bivariate normal integral*, Math. Comp.**32**(1978), no. 141, 277–279. MR**0461849**, https://doi.org/10.1090/S0025-5718-1978-0461849-9**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**, https://doi.org/10.1080/00949659008811236**10.**W.N. Evans, R.M. Schwab, Finishing high school and starting college: do catholic schools make a difference?, J. Econ. 110 (4) (1995), 941-974.**11.**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.**12.**Alan Genz,*Numerical computation of rectangular bivariate and trivariate normal and 𝑡 probabilities*, Stat. Comput.**14**(2004), no. 3, 251–260. MR**2086401**, https://doi.org/10.1023/B:STCO.0000035304.20635.31**13.**Norman L. Johnson and Samuel Kotz,*Distributions in statistics: continuous multivariate distributions*, John Wiley & Sons, Inc., New York-London-Sydney, 1972. Wiley Series in Probability and Mathematical Statistics. MR**0418337****14.**S.H. Martzoukos, The option on assets with exchange rate and exercise price risk. J. Multinatl. Financ. Manage. 11 (1) (2001), 1-15.**15.**C. Nicholson,*The probability integral for two variables*, Biometrika**33**(1943), 59–72. MR**0011409**, https://doi.org/10.1093/biomet/33.1.59**16.**Donald B. Owen,*Tables for computing bivariate normal probabilities*, Ann. Math. Statist.**27**(1956), 1075–1090. MR**0127562**, https://doi.org/10.1214/aoms/1177728074**17.**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.**18.**K. Pearson, Mathematical contributions to the theory of evolution. XI. on the influence of natural selection on the variability and correlation of organs. Philos. Trans. R. Soc. S-A. 200 (1903), 1-66.**19.**W.F. Sheppard, On the calculation of the double-integral expressing normal correlation, Trans. Camb. Philos. Soc. 19 (1900), 23-66.**20.**M.K. Simon, D. Divsalar, Some new twists to problems involving the Gaussian probability integral, IEEE Trans. Commun. 46 (2) (1998), 200-210.**21.**J. Terza, U. Welland, A comparison of bivariate normal algorithms, J. Stat. Comput. Simul. 39 (1991), 115-127.**22.**O.F. Vasicek, A series expansion for the bivariate normal integral, J. Comput. Financ. 1 (1998), 5-10.**23.**Jeffrey M. Wooldridge,*Econometric analysis of cross section and panel data*, 2nd ed., MIT Press, Cambridge, MA, 2010. MR**2768559**

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Additional Information

**Hatem A. Fayed**

Affiliation:
Department of Engineering Mathematics and Physics, Faculty of Engineering, Cairo University, Cairo, Egypt 12613

Email:
h{\textunderscore}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

DOI:
https://doi.org/10.1090/S0025-5718-2013-02720-2

Keywords:
Error function,
normal probability,
Gamma function,
Hermite polynomial,
hypergeometric function,
bivariate normal integral,
tetrachoric series

Received by editor(s):
October 5, 2011

Received by editor(s) in revised form:
February 20, 2012

Published electronically:
May 29, 2013

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.