The Standard BVT Problem

The standard bivariate t distribution function is defined by

$$T_\nu({\bf b}, \rho) = \frac{1}{2\pi\sqrt{1-\rho^2} } \int_... ...2-2\rho x_1x_2}{(1-\rho^2)\nu} )^{-\frac{\nu+2}{2}} dx_2dx_1,$$ (16)

with ${\bf b}= (b_1,b_2)$. An alternate definition (Cornish, 1954) for $T_\nu({\bf b},\rho)$ is

$$T_\nu({\bf b}, \rho) = \frac{2^{1-\nu/2}}{\Gamma(\nu/2)} \in... ...e^{-\frac{s^2}{2}} \Phi(\frac{s}{\sqrt{\nu}}{\bf b},\rho)ds .$$ (17)

The Dunnett and Sobel (1954) BVT algorithm has been carefully implemented by the present author. This algorithm uses finite sums of incomplete beta function values which can easily be computed to accuracies that are at the same level as the underlying accuracy of the implementation (e.g single, double or quadruple precision in Fortran). Motivated by work for the development of TVT algorithms, a new BVT algorithm will be described. The new algorithm uses a generalization of the Plackett formula that is the basis for the BVN algorithms that have already been described. This section will end with a report and discussion of test results for the two algorithms.