Probability Content of Regions Under Spherical Normal Distributions, III: The Bivariate Normal Integral

Abstract

The bivariate normal distribution, with its numerous applications, is of considerable importance and has been studied fairly extensively. Among the first statisticians to investigate the distribution were Sheppard [12] and Karl Pearson [9], the latter from the point of view of his celebrated "tetrachoric functions", which were used as the basis for computing tables of the distribution. Pearson's tables have been extended by the University of California Statistical Laboratory [16] and, more fully, by the National Bureau of Standards [5]. In more recent years, the distribution has been studied among others by Nicholson [6], Polya [10], Cadwell [1] and Owen [7], [8]. Owen has also provided useful tables from which the bivariate normal integral may be evaluated. These tables have been published in [7] and in extended form, together with auxiliary tables, in [8]. (The reader is referred to [8] and [5] for further references and for some interesting applications.) An essential part of the procedures used by Nicholson and Owen is to reduce the integral, which is a function of three parameters, the coordinates $(x_0, y_0)$ of the vertex of the infinite rectangle over which integration is to be extended and the correlation coefficient $\rho$, to functions of only two parameters. The series based on tetrachoric functions for the bivariate normal integral suffers from the disadvantage that it converges rather slowly except when $|\rho|$ is small. The need for an expression which shall be suitable for all values of $\rho$, but more especially for high $|\rho|$, has long been felt (see e.g., David [3]). Formula (3.16), taken in conjunction with (2.7), as well as formula (3.21), is designed to meet this need. These are two-parameter formulae and have the further advantage of being especially useful for high values of $x_0$ and/or $y_0$. Next, the formulae are used to provide equivalent rapidly convergent Stieltjes type continued fractions, known as $S$-fractions (equations (4.6) and (4.19)). These two sets of formulae constitute the basic results of this paper. They are, in fact, analogues of the corresponding known formulae for the univariate normal integral.