Special Cases

Abstract

The problem of evaluating (1.1) or (1.3) can often be simplified by specializing either k, R or a and b. In Section 2.1 we focus on the work that has been done on bivariate and trivariate probabilities and not on general MVN and MVT probabilities. In Section 2.2 we consider calculating probabilities over special integration regions, such orthants, ellipsoids, and hyperboloids. Finally, in Section 2.3 we discuss MVN and MVT problems involving special correlation structures. We do not consider the univariate cases, which have been carefully analyzed elsewhere; see Johnson and Kotz (1970a,b) for extensive discussions and references. Highly accurate implementations for Φ(x), Φ1(x), T(x;), and T1(x;) are available in standard statistical computing environments. We assume the availability of these functions for many of the computational methods that we discuss in this and later chapters.