The Methods

All of the methods that we consider begin with a Cholesky decomposition of $\mbox{\boldmath$\Sigma$}$ in the form $\mbox{\boldmath$\Sigma$}= CC^t$, where $C$ is a lower triangular $m \times m$ matrix. This is followed by the change of variables ${\bf x}= C{\bf y}$, so that ${\bf x}^t \mbox{\boldmath$\Sigma$}^{-1} {\bf x}= {\bf y}^t{\bf y}$, $d{\bf x}=\vert C\vert d{\bf y}= \sqrt{\vert\mbox{\boldmath$\Sigma$}\vert}d{\bf y}$, and therefore

$${\bf T}({\bf a}, {\bf b}, \mbox{\boldmath$\Sigma$}, \nu) = \frac{\Gamma(\frac{\nu+m}{2})} {\Gamma(\frac{\nu}{2})\sqrt{(\nu\p... ...f y}\leq {\bf b}} (1+ \frac{{\bf y}^t{\bf y}}{\nu})^{-\frac{\nu+m}{2}} d{\bf y}$$ (4)

$$\equiv \frac{2^{1-\frac{\nu}{2}}}{\Gamma(\frac{\nu}{2})} \int\limits_0^{... ...y}\leq \frac{s{\bf b}}{\sqrt{\nu}}} e^{-\frac{{\bf y}^t{\bf y}}{2}} d{\bf y}ds,$$ (5)

and

$${\bf T}({\bf a}, {\bf b}, \mbox{\boldmath$\Sigma$}, \nu, \mb... ...boldmath$\delta$}} e^{-\frac{{\bf y}^t{\bf y}}{2}} d{\bf y}ds.$$ (6)

Genz and Bretz (1999) introduced additional transformations for ${\bf T}$ as defined by equation (4). These transformations, which effect a separation of the variables, will be used for some of the methods to be described in the following sections. First, let $K_\nu^{(m)}= \frac{\Gamma(\frac{\nu+m}{2})}{\Gamma(\frac{\nu}{2})(\nu\pi)^{\frac{m}{2}}},$ and notice that $(1+\frac{\sum_{j=1}^my_j^2}{\nu})= (1+\frac{y_1^2}{\nu}) (1+\frac{y_2^2}{\nu+y_1^2})\cdots (1+\frac{y_m^2}{\nu+\sum_{j=1}^{m-1}y_j^2})$. Now let $y_i = u_i\sqrt{\frac{\nu+\sum_{j=1}^{i-1}y_j^2}{\nu+i-1}}$ (which can equivalently be written $y_i = u_i\sqrt{\prod_{j=1}^{i-1}\frac{\nu+j-1+u_j^2}{\nu+j}}$). Then a little algebra (see Genz and Bretz, 1999, for some details) shows that

$${\bf T}({\bf a},{\bf b},\mbox{\boldmath$\Sigma$}, \nu) =\int... ...}^{(1)}}{(1+\frac{u_m^2}{\nu+m-1})^{\frac{m+\nu}{2}}}d{\bf u}.$$ (7)