Abstract
In this paper, a new procedure is described for evaluating the probability that all elements of a normally distributed vector are non-negative, which is called the non-centered orthant probability. This probability is defined by a multivariate integral of the density function. The definition is simple, and this probability arises frequently in statistics because the normal distribution is prevalent. The method for evaluating this probability, however, is not obvious, because applying direct numerical integration is not practical except in low dimensional cases. In the procedure proposed in this paper, the problem is decomposed into sub-problems of lower dimension. Considering the projection onto subspaces, the solutions of the sub-problems can be shared in the evaluation of higher dimensional problems. Thus the sub-problems form a lattice structure. This reduces the computational time from a factorial order, where the interim results are not shared, to order \(p^{2}2^{p}\), which is faster than the procedures that have been reported in the literature.
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Acknowledgments
The author would like to acknowledge Prof. Emer. Hironao Kawashima of Keio University for his help and encouragement, and the editor and referees for kindly providing comments and suggestions. The idea of structuring Sect. 2 as a series of lemmas came from the suggestions and explicit examples provided by the anonymous referees. The extensive lists of comments by the editor and the referees were essential in revising the manuscript. The author also would like to acknowledge Dr. Steven Kraines for his thoughtful English proofreading.
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Nomura, N. Evaluation of Gaussian orthant probabilities based on orthogonal projections to subspaces. Stat Comput 26, 187–197 (2016). https://doi.org/10.1007/s11222-014-9487-8
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DOI: https://doi.org/10.1007/s11222-014-9487-8