Approximations to multivariate $t$ integrals with application to multiple comparison procedures

Abstract

Various multiple comparison procedures involve the evaluation of multivariate normal and $t$ integrals with non-decomposable correlation matrices. While exact methods exist for their computations, it is sometimes necessary to consider simpler and faster approximations. We consider approximations based on approximations to the correlation matrix (methods which provide no error control) as well as inequality based methods (where, by definition, the sign of the error is known). Comparisons of different methods, to assess accuracy, are given for particular multiple comparison problems which require high-dimensional integrations.