Generalized Plackett TVT Algorithms

The formula (24) can be integrated to produce new TVT formulas (analogous to (13-14))

$$T_\nu({\bf b}, R) = T_\nu({\bf b}, R^*) + \frac{1}{2\pi}\in... ...u\big(\frac{u_3(r)}{(1+\frac{f_3(r)}{\nu})^\frac{1}{2}}\big)dr$$ (25)

and

$$T_\nu({\bf b}, R) = T_\nu({\bf b}, R') + \frac{1}{2\pi}\int_{0}^{1}\bigg(\ \rho_{21}\frac{(1+\frac{f_3(\r... ...T_\nu\big(\frac{\hat{u}_3(t)}{(1+\frac{f_3(\rho_{21}t)}{\nu})^\frac{1}{2}}\big)$$

$$\ + \rho_{31}\frac{(1+\frac{f_2(\rho_{31}t)}{\n... ...c{\hat{u}_2(t)}{(1+\frac{f_2(\rho_{31}t)}{\nu})^\frac{1}{2}}\big) \bigg)dt.$$ (26)

The most efficient method for the TVN case used equation (14), so this suggests that equation (26) could be used for an efficient TVT method. Unfortunately, there is no easy computation of $T_\nu({\bf b}, R')$ that uses univariate and bivariate t distribution values. However, a combination of equations (25) and (26) can be used for a practical method. First define

$$R^{**} = \left[ \begin{array}{ccc} 1 & 0& 0 \\ 0 & 1 & s \\ 0 & s & 1 \\ \end{array}\right],$$

with $s = sign(\rho_{32})$. If equation (25) is used, with variables 1 and 3 interchanged, to compute $T_\nu({\bf b}, R')$ from $T_\nu({\bf b}, R^{**})$, then

$$T_\nu({\bf b}, R') = T_\nu({\bf b}, R^{**}) + \frac{1}{2\pi... ...\nu\big(\frac{b_1}{(1+\frac{f_1(r)}{\nu})^\frac{1}{2}}\big)dr,$$

where $f_1(r)=\frac{b_2^2+b_3^2-2rb_2b_3}{(1-r^2)}$. The simplified $b_1$ term in the integrand's $T_\nu$ argument numerator occurs because the zeros in $R^{**}$ result in $u_1(r) = b_1$. The equation for the combined method is

$$T_\nu({\bf b}, R) = T_\nu({\bf b}, R^{**}) + \frac{1}{2\pi}\int_{s}^{\rho_{32}} \frac{(1+\frac{f_1(r)}{\nu})^{... ...}{\sqrt{1-r^2}} T_\nu\big(\frac{b_1}{(1+\frac{f_1(r)}{\nu})^\frac{1}{2}}\big)dr$$

$$+ \frac{1}{2\pi}\int_{0}^{1}\bigg(\ \rho_{21}\frac{(1+\frac{f_3(\r... ...T_\nu\big(\frac{\hat{u}_3(t)}{(1+\frac{f_3(\rho_{21}t)}{\nu})^\frac{1}{2}}\big)$$

$$\ + \rho_{31}\frac{(1+\frac{f_2(\rho_{31}t)}{\n... ...c{\hat{u}_2(t)}{(1+\frac{f_2(\rho_{31}t)}{\nu})^\frac{1}{2}}\big) \bigg)dt.$$ (27)

The singular $T_\nu({\bf b}, R^{**})$ can be efficiently computed with BVT values using

$$T_\nu({\bf b}, R^{**}) = \left\{ \begin{array}{cl} T_\nu((... ...-T_\nu((b_1,-b_3),0)) & \mbox{if s = -1} \end{array} \right. ,$$

where $T_\nu((b_1,b_2),\rho)$ is the standard bivariate t distribution function.