A Generalized Plackett TVT Formula

A generalization of Plackett's formula requires ${\partial T_\nu({\bf b}, R)}/{\partial\rho_{21}}$. If definition (23) is used for $T_\nu({\bf b}, R)$, then

$$\frac{\partial T_\nu({\bf b}, R)}{\partial\rho_{21}}= \frac... ...ial\Phi(\frac{s}{\sqrt{\nu}}{\bf b},R)}{\partial\rho_{21}} ds,$$

so Plackett's TVN formula (12) can be applied, giving

$$\frac{\partial T_\nu({\bf b}, R)}{\partial\rho_{21}} = \frac... ...infty}^{\frac{s}{\nu} u_3(\rho_{21})} e^{-\frac{x^2}{2}}dxds.$$

If the inner integral is transformed using $x = \frac{s}{\sqrt{\nu}}y$, the exponential terms are combined, a second change of variables $r=s(1+\frac{f_3(\rho_{21})+y^2}{\nu})^\frac{1}{2}$ is used, and the order of integration is reversed, the result is

$$\frac{\partial T_\nu({\bf b}, R)}{\partial\rho_{21}}= \frac... ...rho_{21}^2}} \int_{0}^{\infty}r^{\nu}e^{-\frac{r^2}{2}} drdy.$$

Next, the factorization $(1+\frac{f_3(\rho)+y^2}{\nu})= (1+\frac{f_3(\rho)}{\nu})(1+\frac{y^2}{\nu+f_3(\rho)})$ is used, along with the inner integral value $\Gamma(\frac{\nu+1}{2})2^{\frac{\nu+1}{2}-1}$, and some algebra, to obtain

$$\frac{\partial T_\nu({\bf b}, R)}{\partial\rho_{21}}= \frac... ...1})} (1+\frac{y^2}{\nu+f_3(\rho_{21})})^{-\frac{\nu+1}{2}}dy.$$

The final change of variables, $z = y/(1+\frac{f_3(\rho_{21})}{\nu})^\frac{1}{2}$, and some more algebra, produces the new trivariate t formula

$$\frac{\partial T_\nu({\bf b}, R)}{\partial\rho_{21}}= \frac... ...}{\nu})^\frac{1}{2}} (1+\frac{z^2}{\nu})^{-\frac{\nu+1}{2}}dz.$$

This can be written in a form very similar to the form for equation (12) as

$$\frac{\partial T_\nu({\bf b}, R)}{\partial\rho_{21}}= \frac... ...(\rho_{21})}{(1+\frac{f_3(\rho_{21})}{\nu})^\frac{1}{2}}\big).$$ (24)

Appropriate permutations of the $b$'s and the $\rho$'s can be used to provide similar formulas for $\partial T_\nu({\bf b}, R)/\partial\rho_{31}$ and $\partial T_\nu({\bf b}, R)/\partial\rho_{32}$.