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


Authors: Hatem A. Fayed and Amir F. Atiya
Journal: Math. Comp. 83 (2014), 2385-2402
MSC (2010): Primary 42A16, 62H86
DOI: https://doi.org/10.1090/S0025-5718-2014-02844-5
Published electronically: April 17, 2014
MathSciNet review: 3223336
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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.


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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/S0025-5718-2014-02844-5
Keywords: Multivariate normal probability integral, Fourier series, tetrachoric series
Received by editor(s): June 28, 2012
Received by editor(s) in revised form: January 9, 2013
Published electronically: April 17, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.