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Calculation of orthant probabilities by the holonomic gradient method

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Abstract

We apply the holonomic gradient method (Nakayama et al. Adv Appl Math 47:639–658, 2011) to the calculation of the probabilities of the multivariate normal distribution. The holonomic gradient method applied to finding the orthant probabilities is found to be a variant of Plackett’s recurrence relation. However, an implementation of the method yields recurrence relations that are more suitable for numerical computation than is Plackett’s recurrence relation. We derive some theoretical results on the holonomic system for the orthant probabilities. These results show that multivariate normal orthant probabilities possess some remarkable properties from the viewpoint of holonomic systems. Finally, we show that the numerical performance of our method is comparable or superior to that of existing methods.

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Acknowledgments

The first author is a research fellow of Japan Society for the Promotion of Science. His work is supported by Grant-in-Aid for JSPS Fellows 26.3125. The second author is supported by JSPS Grant-in-Aid for Scientific Research No. 25220001.

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Authors and Affiliations

  1. Graduate School of Information Science and Technology, University of Tokyo, Bunkyo, Tokyo, 113-8656, Japan

    Tamio Koyama

  2. Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Bunkyo, Tokyo, 113-8656, Japan

    Akimichi Takemura

Authors
  1. Tamio Koyama
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  2. Akimichi Takemura
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Correspondence to Tamio Koyama.

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Koyama, T., Takemura, A. Calculation of orthant probabilities by the holonomic gradient method. Japan J. Indust. Appl. Math. 32, 187–204 (2015). https://doi.org/10.1007/s13160-015-0166-8

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  • DOI: https://doi.org/10.1007/s13160-015-0166-8

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