MATHEMATICS OF COMPUTATION Volume 83, Number 289, September 2014, Pages 2385–2402 S 0025-5718(2014)02844-5 Article electronically published on April 17, 2014 A NOVEL SERIES EXPANSION FOR THE MULTIVARIATE NORMAL PROBABILITY INTEGRALS BASED ON FOURIER SERIES HATEM A. FAYED AND AMIR F. ATIYA 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 ∞ to be from -∞ to ∞ 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. 1. Introduction A common problem that arises in many statistics computations is computing the complementary integral of the multivariate normal distribution which is given by L(h, Σ) = 1  (2π) m |Σ| ∞  h 1 ··· ∞  h m exp  − 1 2 x T Σ −1 x  dx, where x =(x 1 ,x 2 , ··· ,x m ) and Σ is an m×m symmetric positive definite covariance matrix. The problem has received considerable attention in the literature [14–16, 18]. Efficient formulas exist for m = 1 and m = 2. For m = 2, Pearson [21] derived the following simple tetrachoric series for the bivariate normal CDF: Φ(h 1 ,h 2 ,ρ) = Φ(h)Φ(k)+ 1 2π exp  − (h 2 1 + h 2 2 ) 2  ∞  j=0 2 −j ρ j+1 j + 1! H j ( h 1 √ 2 )H j ( h 2 √ 2 ), where L(h 1 ,h 2 ,ρ) = Φ(−h 1 , −h 2 ,ρ) and H j (x) is the Hermite polynomial defined by Abramowitz and Stegun [1, p. 775]: H j (x)=(−1) j exp(x 2 )D j exp(−x 2 )= j ! [j/2]  k=0 (−1) k k!(j − 2k)! (2x) j−2k [j /2] =  j /2 j even, (j − 1)/2 j odd, Received by the editor June 28, 2012 and, in revised form, January 9, 2013. 2010 Mathematics Subject Classification. Primary 42A16, 62H86. Key words and phrases. Multivariate normal probability integral, Fourier series, tetrachoric series. c 2014 American Mathematical Society Reverts to public domain 28 years from publication 2385 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use