Orthant probabilities of elliptical distributions from orthogonal projections to subspaces

摘要

A new procedure is proposed for evaluating non-centred orthant probabilities of elliptical distributed vectors, which is the probabilities that all elements of a vector are non-negative. The definition of orthant probabilities is simple, formulated as a multiple integral of the density function; however, applying direct numerical integration is not practical, except in low-dimensional cases, and methods for evaluating orthant probabilities are not trivial. This probability arises frequently in statistics; in particular, the normal distribution and Student's t-distribution are in the family of elliptical distribution. In the procedure proposed in this paper, an orthant probability is approximated by the probability that the vector falls in a simplex. In the process, the problem is decomposed into sub-problems of lower dimension based on the symmetry of elliptical distributions. Intermediate sub-problems can be generated by projection onto subspaces, and the sub-problems form a lattice structure. Considering this structure, intermediate computations are shared between the evaluations of higher-dimensional problems, and computational time is reduced. The procedure can be applied not only to normal distributions, but also to general elliptical distributions, especially t-distributions, which are used in the multiple comparison procedure.