Calculation of orthant probabilities by the holonomic gradient method

TL;DR

This paper introduces a novel application of the holonomic gradient method to efficiently compute multivariate normal orthant probabilities, offering improved numerical performance over existing techniques.

Contribution

The paper develops a holonomic gradient method tailored for orthant probabilities, providing theoretical insights and demonstrating superior numerical efficiency.

Findings

The method is comparable or superior in performance to existing methods.

Orthant probabilities exhibit special properties within holonomic systems.

The approach offers a numerically stable and efficient computation technique.

Abstract

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.