Conclusions

Algorithms for accurate and efficient computations of BVN, TVN, BVT and TVT probabilities were considered. Generalizations of Plackett's Normal formulas were derived for the bivariate t and trivariate t cases. An implementation of an algorithm using the generalized bivariate t formula was significantly slower than an implementation of the Dunnett and Sobel (1954) algorithm. The Dunnett and Sobel algorithm continues to be the most efficient algorithm for BVT computations. Although Plackett formula algorithms for TVN and TVT probabilities are susceptible to moderate loss of accuracy for some types of problems, implementations of algorithms that use adaptive integration can efficiently produce double precision results for most problems, with times that are less than $O(10^{-4})$ seconds for Fortran implementations running on modern computer workstations. Fortran software for all of the algorithms discussed in this paper is available from the author's website.