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1 results

  • On spherical Monte Carlo simulations for multivariate normal probabilities

  • Part of
  • Huei-Wen Teng, Ming-Hsuan Kang, Cheng-Der Fuh
  • Journal:
    Advances in Applied Probability / Volume 47 / Issue 3 / September 2015
    Published online by Cambridge University Press:
    21 March 2016, pp. 817-836
    Print publication:
    September 2015
    • Article
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  • The calculation of multivariate normal probabilities is of great importance in many statistical and economic applications. In this paper we propose a spherical Monte Carlo method with both theoretical analysis and numerical simulation. We start by writing the multivariate normal probability via an inner radial integral and an outer spherical integral using the spherical transformation. For the outer spherical integral, we apply an integration rule by randomly rotating a predetermined set of well-located points. To find the desired set, we derive an upper bound for the variance of the Monte Carlo estimator and propose a set which is related to the kissing number problem in sphere packings. For the inner radial integral, we employ the idea of antithetic variates and identify certain conditions so that variance reduction is guaranteed. Extensive Monte Carlo simulations on some probabilities confirm these claims.