A representation of multivariate normal probability integrals by integral transforms

Abstract

By using the integral representation of the Hermite polynomial, the tetrachoric series for arbitrary dimension K is transformed into a finite sum of multivariate Fourier transforms over (−∞, ∞) each involving the normal characteristic function divided by the produot of the variables of integration. These integral transforms are evaluated numerically by Gaussian quadrature and provide rapidly convergent formulae for the multivariate normal probability integral.