Calculation of orthant probabilities by the holonomic gradient method

We apply the holonomic gradient method (HGM) introduced by [9] to the calculation of orthant probabilities of multivariate normal distribution. The holonomic gradient method applied to orthant probabilities is found to be a variant of Plackett's recurrence relation ([14]). However an implementation of the method yields recurrence relations more suitable for numerical computation than Plackett's recurrence relation. We derive some theoretical results on the holonomic system for the orthant probabilities. These results show that multivariate normal orthant probabilities possess some remarkable properties from the viewpoint of holonomic systems. Finally we show that numerical performance of our method is comparable or superior compared to existing methods.