Abstract
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.
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The author would like to express acknowledgement to the editor and referees for their kindly comments and suggestions. In particular, one of the anonymous referees gave innumerable suggestions and many points to correct in the original manuscript. The suggestions were significant in the process of publication of this paper.
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Nomura, N. Orthant probabilities of elliptical distributions from orthogonal projections to subspaces. Stat Comput 29, 289–300 (2019). https://doi.org/10.1007/s11222-018-9808-4
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DOI: https://doi.org/10.1007/s11222-018-9808-4