Plackett TVN Formulas

The formula derived by Plackett for a correlation coefficient partial derivative of the trivariate normal distribution can be written

$$\frac{\partial \Phi({\bf b}, R)}{\partial\rho_{21}} = \frac... ...rho_{21})/2}} {2\pi\sqrt {1-\rho_{21}^2}}\Phi(u_3(\rho_{21})),$$ (12)

where $f_3(r)=\frac{b_1^2+b_2^2-2rb_1b_2}{(1-r^2)}$ and

$$u_3(r) = \frac{b_3(1-r^2)-b_1(\rho_{31}-r\rho_{32})-b_2(\rho... ...^2-\rho_{31}^2-\rho_{32}^2+2r\rho_{31}\rho_{32}))^\frac{1}{2}}.$$

Formulas for the two other off-diagonal $\rho_{ij}$ partial derivatives can be obtained from equation (12) with appropriate permutations of the $b$'s and $\rho$'s.

Integration of this formula can provide TVN formulas which involve only two-dimensional integrals. Two Plackett formula methods, studied by Gassmann (2000), are also considered here. The first method uses

$$\Phi({\bf b}, R) = \Phi({\bf b}, R^*) + \frac{1}{2\pi}\int_{... ...\rho_{21}} \frac{e^{-f_3(r)/2}} {\sqrt {1-r^2}}\Phi(u_3(r))dr,$$ (13)

where $\rho_{21}^*=\rho_{31}\rho_{32}\pm\sqrt{(1-\rho_{31}^2)(1-\rho_{32}^2)}$ and

$$R^* = \left[ \begin{array}{ccc} 1 & \rho_{21}^* & \rho_{31... ..._{32} \\ \rho_{31} & \rho_{32} & 1 \\ \end{array}\right] .$$

The matrix $R^*$ is singular, so $\Phi({\bf b}, R^*)$ can be computed using univariate and bivariate distribution values only. For efficient computation, a permutation of the $\rho$'s in $R$ (along with corresponding $b$'s), and the sign used for $\rho_{21}^*$, are chosen to minimize the integration interval width $\vert\rho_{21}^*-\rho_{21}\vert$.

The second Plackett formula method uses

$$\Phi({\bf b}, R) = \Phi({\bf b}, R') +\frac{1}{2\pi}\int_{0... ...}t)/2}}{\sqrt{1-{\rho_{31}^2t^2}}}\Phi(\hat{u}_2(t)) \bigg)dt,$$ (14)

where $f_2(r)=\frac{b_1^2+b_3^2-2rb_1b_3}{(1-r^2)}$,

$$R' = \left[ \begin{array}{ccc} 1 & 0& 0 \\ 0 & 1 & \rho_{32} \\ 0 & \rho_{32} & 1 \\ \end{array}\right] ,$$

$$\hat{u}_2(t) = \frac{b_2(1-\rho_{31}^2t^2)-b_1t(\rho_{21}-\rh... ...\rho_{32}^2 + 2t^2\rho_{31}\rho_{21}\rho_{32}))^\frac{1}{2}},$$

and

$$\hat{u}_3(t) = \frac{b_3(1-\rho_{21}^2t^2)-b_1t(\rho_{31}-\rh... ...-\rho_{32}^2 + 2t^2\rho_{31}\rho_{21}\rho_{32}))^\frac{1}{2}}.$$

In this case, $\Phi({\bf b}, R')$ is easily computed as $\Phi(b_1)\Phi((b_2,b_3),\rho_{32})$. For efficient numerical integration, a permutation of the $\rho$'s and $b$'s, is chosen to minimize $max(\vert\rho_{31}\vert,\vert\rho_{21}\vert)$. The integrals for both Plackett formula methods were transformed using $r = sin(\theta)$, in order to remove integrand denominator singularities. This second Plackett method (14) was the method that Drezner (1994) implemented.