The Evaluation of Multi-normal Non-centred Orthant Probabilities

• Activity Number: 33
• Type: Topic Contributed
• Date/Time: Sunday, August 11, 2002 : 4:00 PM to 5:50 PM
• Sponsor: Section on Statistical Computing*
• Author(s): Tetsuhisa Miwa*+ and Anthony Hayter and Satoshi Kuriki
• Affiliation(s): National Institute for Agro-Environmental Sciences and Georgia Institute of Technology and The Institute of Statistical Mathematics
• Keywords: cumulative distribution function ; orthoscheme probabilities ; polyhedral cone ; recursive integration

Abstract

We propose an efficient procedure to calculate a non-centred orthant probability that all the elements of a multi-variate normal random variable are positive. The multivariate normal distribution can have any positive-definite correlation matrix and any mean vector.

The approach has two stages. In the first stage, we consider non-centred orthoscheme probabilities with tridiagonal correlation matrices. The original multi-variate variable can be transformed to a new variable such that the desired probability is written as a sequence of one-dimensional integral expressions. These integral expressions are evaluated by a recursive computational approach. Then the computational time increases only linearly in the dimensionality, and therefore sufficient accuracy can be achieved.

In the second stage, some ideas of Schlaefli (1858) and Abrahamson (1964) are extended to show that any non-centred orthant probability can be expressed as differences between at most (m-1)! non-centred orthoscheme probabilities, where m is the dimensionality.