Bivariate and trivariate probability distribution computations are needed for many statistics applications. Although reliable, efficient and accurate algorithms for univariate probabilities have been available for some time, high quality algorithms for bivariate and trivariate probability distribution computations have only more recently started to become available. There is a need for these algorithms as components in statistical computation libraries and packages. There is also an increasing need for these algorithms as a means for efficiently computing good bounds for multivariate probabilities. These bounds can then be used to provide more efficient computation methods for multivariate probability computations (see Gassmann, 2000, and Genz, Bretz, and Hochberg, 2003).
There are now many algorithms available for computation of bivariate normal (BVN) probabilities, but the quality of these algorithms has significant variation. In a comparative study of these algorithms by Terza and Welland (1990), Divgi's method (1979) was found to outperform a number of other methods for low accuracy work. Shortly after that paper appeared, Drezner and Wesolowsky (1990) presented a simple method for single precision work that used less computation time than Divgi's algorithm. Recently, after a study of different algorithms, Patefield and Tandy (2000) developed a hybrid double precision algorithm.
Until very recently, there were no accurate and efficient algorithms available for the general trivariate normal (TVN) problem. Schervish's algorithm (1984) was the first published numerical algorithm, but this algorithm was developed for the general multivariate normal problem. This algorithm was followed by algorithms developed by Cox and Wermuth (1991), Wang and Kennedy (1992), and Drezner (1992, 1994). The recent paper by Gassmann (2000) studied these algorithms and made some recommendations for algorithms for efficient and reliable TVN computations. The primary algorithm for bivariate t (BVT) probabilities was developed by Dunnett and Sobel (1954). There are several general algorithms available for multivariate t probability computations (see the review by Genz and Bretz, 2002) but these algorithms usually cannot efficiently provide high accuracy results. There are currently no published specialized algorithms for trivariate t (TVT) probabilities.
In this paper there is a brief discussion of some modifications to the algorithm of Drezner and Wesolowsky which provide a double precision BVN algorithm in a simpler form than the algorithm provided by Patefield and Tandy (2000). This is used as the basis for some TVN algorithms, and some comparisons of different TVN algorithms are given, along with some discussion of extensions of Gassmann's work for reliable double precision computations. These considerations provide some background for work on algorithms for bivariate and trivariate t probability computations. The main contributions of this paper are some generalizations of Plackett's (1954) formulas which provide the basis for some new algorithms for BVT and TVT probabilities. There is a discussion of test results for the new algorithms. Fortran software for BVN, BVT, TVN and TVT double precision implementations is available from the author's website (www.math.wsu.edu/faculty/genz/homepage, in TVPACK).