A Generalized Plackett BVT Formula

A bivariate t generalization of Plackett's formula requires ${\partial T_\nu({\bf b}, \rho)}/{\partial\rho}$. If definition (17) is used for $T_\nu({\bf b},\rho)$, and Plackett's formula (2) is used, then

$$\frac{\partial T_\nu({\bf b},\rho)}{\partial \rho} = \frac{2... ...rac{e^{-\frac{s^2f_3(\rho)}{2\nu}}} {2\pi\sqrt {1-\rho^2}}ds.$$

If the exponential terms are combined, and this is followed by the change of variables $r=s(1+\frac{f_3(\rho_{21})}{\nu})^\frac{1}{2}$, the result is

$$\frac{\partial T_\nu({\bf b},\rho)}{\partial\rho}=\frac{2^{1... ...rt {1-\rho^2}} \int_0^{\infty}r^{\nu-1} e^{\frac{-r^2}{2}}dr.$$

The integral value is $\Gamma(\frac{\nu}{2})2^{\frac{\nu}{2}-1}$, so after expanding $f_3(\rho)$, the bivariate t generalization of Plackett's formula is given by

$$\frac{\partial T_\nu({\bf b}, \rho)}{\partial\rho}= \frac{(... ...1b_2}{\nu(1-\rho^2)})^{-\frac{\nu}{2}}} {2\pi\sqrt{1-\rho^2}}.$$ (18)

The new formula (18) can be integrated to produce new BVT formulas

$$T_\nu({\bf b}, \rho) = T_\nu({\bf b},0) + \frac{1}{2\pi}\in... ..._2^2-2rb_1b_2}{\nu(1-r^2})^{-\frac{\nu}{2}}} {\sqrt{1-r^2}}dr,$$ (19)

and

$$T_\nu({\bf b}, \rho) = T_\nu({\bf b},s) + \frac{1}{2\pi}\in... ...2^2-2rb_1b_2}{\nu(1-r^2)})^{-\frac{\nu}{2}}} {\sqrt{1-r^2}}dr,$$ (20)

where, $s = sign(\rho)$ and

$$T_\nu({\bf b}, s) = \left\{ \begin{array}{cl} T_\nu(min(b_... ...\nu(b_1)-T_\nu(-b_2)) & \mbox{if s = -1} \end{array} \right. ,$$

with $T_\nu(b)$ defined as the standard univariate Student's t distribution. In contrast to the normal case, there is no easy computation of $T_\nu({\bf b}, 0)$ that uses a product of univariate t distribution values. Therefore, a numerical implementation of an algorithm based on equation (20) was developed. After the change of variables $r = sin(\theta)$

$$T_\nu({\bf b}, \rho) = T_\nu({\bf b},s) + \frac{1}{2\pi}\int... ...a)b_1b_2}{\nu \cos(\theta)^2}\bigg) ^{-\frac{\nu}{2}}d\theta ,$$ (21)

numerical integration can be used to approximate the integral.