Introduction

A common problem in many statistics applications is the numerical computation of the multivariate t (MVT) distribution function (see Tong, 1990) defined by

$${\bf T}({\bf a}, {\bf b}, \mbox{\boldmath$\Sigma$}, \nu) = \frac{\Gamma(\frac{\nu+m}{2})} {\Gamma(\frac{\nu}{2})\sqrt{\vert\... ...f x}^t \mbox{\boldmath$\Sigma$}^{-1} {\bf x}}{\nu})^{-\frac{\nu+m}{2}} d{\bf x}$$ (1)

$$\equiv \frac{2^{1-\frac{\nu}{2}}}{\Gamma(\frac{\nu}{2})} \int\limits_0^{... ...s{\bf a}}{\sqrt{\nu}},\frac{s{\bf b}}{\sqrt{\nu}},\mbox{\boldmath$\Sigma$}) ds.$$ (2)

The second form for the MVT distribution function (Cornish, 1954) uses the multivariate Normal (MVN) distribution function, defined by

$$\mbox{\boldmath $\Phi$}({\bf a},{\bf b}, \mbox{\boldmath $\S... ...2} {\bf x}^t \mbox{\boldmath $\Sigma$}^{-1} {\bf x}} d{\bf x}.$$

This definition of the MVT distribution function is also used in the definition of the non-central MVT (NCMVT),

$${\bf T}({\bf a}, {\bf b}, \mbox{\boldmath$\Sigma$}, \nu, \mb... ...t{\nu}}-\mbox{\boldmath$\delta$},\mbox{\boldmath$\Sigma$}) ds.$$ (3)

In all of these definitions ${\bf x}= (x_1, x_2, ..., x_m)^t$, $\mbox{\boldmath$\Sigma$}$ is an $m \times m$ symmetric positive definite covariance matrix and $-\infty \leq a_i<b_i \leq \infty$, for $i = 1, \ldots, m$. In the NCMVT case, the non-centrality vector $\mbox{\boldmath$\delta$}$ has components that satisfy $-\infty < \delta_i < \infty$. The purpose of this paper is to compare different methods for the numerical computation of MVT probabilities. We also discuss some methods for NCMVT probabilities.

There is reliable and efficient software available for computing ${\bf T}$ for $m = 1$, so we assume $m > 1$. The simplest traditional methods use acceptance-rejection sampling. Other methods for $m > 1$ use algorithms developed by Somerville (1997, 1998 and 1999), and Genz and Bretz (1999). We consider acceptance-rejection sampling, Somerville and related methods, the method of Genz and Bretz, and other methods that have not been carefully considered for MVT and NCMVT problems. In Section 2 we provide brief descriptions for various methods, in Section 3 we describe algorithms for implementing the methods and we report test results for the methods.