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The holonomic gradient method for the distribution function of the largest root of a Wishart matrix

Author

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  • Hashiguchi, Hiroki
  • Numata, Yasuhide
  • Takayama, Nobuki
  • Takemura, Akimichi

Abstract

We apply the holonomic gradient method introduced by Nakayama et al. (2011) [23] to the evaluation of the exact distribution function of the largest root of a Wishart matrix, which involves a hypergeometric function 1F1 of a matrix argument. Numerical evaluation of the hypergeometric function has been one of the longstanding problems in multivariate distribution theory. The holonomic gradient method offers a totally new approach, which is complementary to the infinite series expansion around the origin in terms of zonal polynomials. It allows us to move away from the origin by the use of partial differential equations satisfied by the hypergeometric function. From the numerical viewpoint we show that the method works well up to dimension 10. From the theoretical viewpoint the method offers many challenging problems both to statistics and D-module theory.

Suggested Citation

  • Hashiguchi, Hiroki & Numata, Yasuhide & Takayama, Nobuki & Takemura, Akimichi, 2013. "The holonomic gradient method for the distribution function of the largest root of a Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 296-312.
    • Handle: RePEc:eee:jmvana:v:117:y:2013:i:c:p:296-312
    DOI: 10.1016/j.jmva.2013.03.011
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    References listed on IDEAS

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    1. Takemura, Akimichi & Sheena, Yo, 2005. "Distribution of eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues are infinitely dispersed and its application to minimax estimation of covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 94(2), pages 271-299, June.
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    Cited by:

    1. Takayama, Nobuki & Jiu, Lin & Kuriki, Satoshi & Zhang, Yi, 2020. "Computation of the expected Euler characteristic for the largest eigenvalue of a real non-central Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    2. Shimizu, Koki & Hashiguchi, Hiroki, 2021. "Heterogeneous hypergeometric functions with two matrix arguments and the exact distribution of the largest eigenvalue of a singular beta-Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    3. Daya K. Nagar & Raúl Alejandro Morán-Vásquez & Arjun K. Gupta, 2015. "Extended Matrix Variate Hypergeometric Functions and Matrix Variate Distributions," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2015, pages 1-15, January.
    4. Hashiguchi, Hiroki & Takayama, Nobuki & Takemura, Akimichi, 2018. "Distribution of the ratio of two Wishart matrices and cumulative probability evaluation by the holonomic gradient method," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 270-278.
    5. Tamio Koyama & Hiromasa Nakayama & Kenta Nishiyama & Nobuki Takayama, 2014. "Holonomic gradient descent for the Fisher–Bingham distribution on the $$d$$ d -dimensional sphere," Computational Statistics, Springer, vol. 29(3), pages 661-683, June.

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