A Simple Approximation for Bivariate and Trivariate Normal Integrals

Abstract

Summary A simple approximation for the bivariate normal distribution function is described, together with a second-order refinement. For Ip| < 0-9, the worst error is about 10% arising when both arguments of the distribution function are equal, but over most of the range the agreement is much closer. An extension to trivariate normal integrals has similar good properties. Numerical evaluation of the bivariate normal distribution function is required for a number of probabilistic and statistical purposes and there is no general closed form expression. The computer algorithm of Donnelly (1973) and its extension to more dimensions by Schervish (1984) are available for accurate numerical evaluation and have been extensively used in the present study. The National Buteau of Standards tables (1959) are comprehensive and easily used, especially for 'simple' values of the correlation coefficient. Owen (1956) discusses the numerical analytical aspects, provides the theoretical basis of Donnelly's algorithms and gives a concise form of table. It is still helpful, however, to have an explicit formula for at least three reasons: to aid further analytical development, to be employed in rapid 'pocket calculator'-based evaluation, especially for preliminary calculations, and for computerized use when a large number of evaluations are required and speed of computation is important. The present note gives such a formula.