The Standard TVT Problem

The standard trivariate t distribution function is defined by

$$T_\nu({\bf b}, R) = \frac{\Gamma((\nu+3)/2)}{\Gamma(\nu/2) ... ...\bf x}^TR^{-1}{\bf x}}{\nu} )^{-\frac{\nu+3}{2}} dx_3dx_2dx_1,$$ (22)

where $R$ is a correlation matrix. An alternate definition (Cornish, 1954) is

$$T_\nu({\bf b}, R) = \frac{2^{1-\nu/2}}{\Gamma(\nu/2)} \int_{... ...-1}e^{-\frac{s^2}{2}} \Phi(\frac{s}{\sqrt{\nu}}{\bf b},R)ds .$$ (23)

In order to develop an efficient TVT algorithm, some experimentation was initially done with equation (23) using algorithms that combine a numerical integration method for the outer integral with an efficient method for the inner TVN integrals. Some of the results of this experimentation will be discussed later in this section, but these methods, which use the four-dimensional integral in equation (23), do not appear to be as efficient as methods that use a generalization of Plackett's formula, and will not be considered in any detail here .