Name: Miwa

Firstname: Tetsuhisa

Title: Dr

Institution: National Institute for Agro-Environmental Sciences

Street: 3-1-3 Kannondai

City: Tsukuba

Zip-Code: 305-8604

Country: Japan

Phone: +81-298-38-8224

Fax: +81-298-38-8199

Email: miwa@niaes.affrc.go.jp

Authors: Tetsuhisa Miwa (National Institute for Agro-Environmental Sciences) Tony Hayter (Georgia Institute of Technology) Satoshi Kuriki (The Institute of Statistical Mathematics)

Title: The efficient evaluation of multi-normal distribution functions

Abstract: We propose an efficient procedure for evaluating the cumulative distribution function of a multivariate normal distribution. The multivariate normal distribution can have any positive-definite correlation matrix and any non-zero mean vector.

The approach has two stages. In the first stage, we consider non-centred orthoscheme probabilities (When the correlation matrix is tri-diagonal, the corresponding multi-normal distribution function is called a non-centred orthoscheme probability). The original multivariate 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 multi-normal distribution function can be expressed as differences between at most $(m-1)!$ non-centred orthoscheme probabilities, where $m$ is the dimensionality.

This approach therefore allows the accurate evaluation of the multi-normal distribution function for any correlation matrix and any non-zero mean vector.

Key words: orthant probability; orthoscheme probability; polyhedral cone; recursive integration

References: Abrahamson, I. G. (1964) Orthant probabilities for the quadrivariate normal distribution. {\it Ann. Math. Statist.}, {\bf 35}, 1685-1703.

Miwa, T., Hayter, A. J. and Liu, W. (2000) Calculations of level probabilities for normal random variables with unequal variances with applications to Bartholomew`s test in unbalanced one-way models. {\it Comput. Statist. Data Anal.}, {\bf 34}, 17--32.

Schlaefli, L. (1858) On the multiple integral {$\int^n dxdy\ldots dz$}, whose limits are $p_1=a_1x+b_1y+\cdots+h_1z>0,\ p_2>0,\ \ldots,\ p_n>0$, and $x^2+y^2+\cdots+z^2<1$. {\it Quart. J. Pure Appl. Math.}, {\bf 2}, 269-301.