Computation of multivariate normal probabilities with polar coordinate systems

Abstract

We consider the problem of evaluation of the probability that all elements of a multivariate normally distributed vector have non-negative coordinates; this probability is called the non-centred orthant probability. The necessity for the evaluation of this probability arises frequently in statistics. The probability is defined by the integral of the probability density function. However, direct numerical integration is not practical. In this article, a method is proposed for the computation of the probability. The method involves the evaluation of a measure on a unit sphere surface in p-dimensional space that satisfies conditions derived from a covariance matrix. The required computational time for the p-dimensional problem is proportional to p²·2^p−1, and it increases at a rate that is lower than that in the case of the existing method.

Keywords:

• cumulative distribution function
• polyhedral cone
• orthant probability
• polar coordinates
• recursive integration

AMS Subject Classifications:

• 65C60
• 62H99
• 60E99

Acknowledgements

The author thanks professor Hironao Kawashima at Keio University for his help and constant encouragement. The author also likes to express his sincere thanks for the comments from the anonymous reviewers and the anonymous associate editor; their comments were indispensable in improving the manuscript.