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The evaluation of general non‐centred orthant probabilities

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  • Tetsuhisa Miwa
  • A. J. Hayter
  • Satoshi Kuriki

Abstract

Summary. The evaluation of the cumulative distribution function of a multivariate normal distribution is considered. The multivariate normal distribution can have any positive definite correlation matrix and any mean vector. The approach taken has two stages. In the first stage, it is shown how non‐centred orthoscheme probabilities can be evaluated by using a recursive integration method. In the second stage, some ideas of Schläfli and Abrahamson are extended to show that any non‐centred orthant probability can be expressed as differences between at most (m−1)! non‐centred orthoscheme probabilities. This approach allows an accurate evaluation of many multivariate normal probabilities which have important applications in statistical practice.

Suggested Citation

  • Tetsuhisa Miwa & A. J. Hayter & Satoshi Kuriki, 2003. "The evaluation of general non‐centred orthant probabilities," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(1), pages 223-234, February.
    • Handle: RePEc:bla:jorssb:v:65:y:2003:i:1:p:223-234
    DOI: 10.1111/1467-9868.00382
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    Cited by:

    1. Z. I. Botev, 2017. "The normal law under linear restrictions: simulation and estimation via minimax tilting," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(1), pages 125-148, January.
    2. Lee, Hangsuck & Ha, Hongjun & Kong, Byungdoo, 2024. "Pricing first-touch digitals with a multi-step double boundary and American barrier options," Finance Research Letters, Elsevier, vol. 59(C).
    3. Phinikettos, Ioannis & Gandy, Axel, 2011. "Fast computation of high-dimensional multivariate normal probabilities," Computational Statistics & Data Analysis, Elsevier, vol. 55(4), pages 1521-1529, April.
    4. Jafari Khaledi, Majid & Zareifard, Hamid & Boojari, Hossein, 2023. "A spatial skew-Gaussian process with a specified covariance function," Statistics & Probability Letters, Elsevier, vol. 192(C).
    5. Madar, Vered, 2015. "Direct formulation to Cholesky decomposition of a general nonsingular correlation matrix," Statistics & Probability Letters, Elsevier, vol. 103(C), pages 142-147.
    6. A. Hayter & Y. Lin, 2012. "The evaluation of two-sided orthant probabilities for a quadrivariate normal distribution," Computational Statistics, Springer, vol. 27(3), pages 459-471, September.
    7. Lee, Hangsuck & Ha, Hongjun & Kong, Byungdoo & Lee, Minha, 2023. "Pricing multi-step double barrier options by the efficient non-crossing probability," Finance Research Letters, Elsevier, vol. 54(C).
    8. Paul Kabaila & Nishika Ranathunga, 2021. "Computation of the expected value of a function of a chi-distributed random variable," Computational Statistics, Springer, vol. 36(1), pages 313-332, March.
    9. Jietao Xie & Juan Wu, 2020. "Recursive Calculation Model for a Special Multivariate Normal Probability of First-Order Stationary Sequence," INFORMS Journal on Computing, INFORMS, vol. 32(1), pages 164-171, January.
    10. Jacques Huguenin & Florian Pelgrin & Alberto Holly, 2009. "Estimation of multivariate probit models by exact maximum likelihood," Working Papers 0902, University of Lausanne, Institute of Health Economics and Management (IEMS).
    11. Barry C. Arnold & Robert J. Beaver, 2007. "Skewing Around: Relationships Among Classes of Skewed Distributions," Methodology and Computing in Applied Probability, Springer, vol. 9(2), pages 153-162, June.

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