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Heterogeneous hypergeometric functions with two matrix arguments and the exact distribution of the largest eigenvalue of a singular beta-Wishart matrix

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  • Shimizu, Koki
  • Hashiguchi, Hiroki

Abstract

This paper discusses certain properties of heterogeneous hypergeometric functions with two matrix arguments. These functions are newly defined but have already appeared in statistical literature and are useful when dealing with the derivation of certain distributions for the eigenvalues of singular beta-Wishart matrices. The joint density function of the eigenvalues and the distribution of the largest eigenvalue can be expressed in terms of heterogeneous hypergeometric functions. Exact computation of the distribution of the largest eigenvalue is conducted for real and complex cases.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:jmvana:v:183:y:2021:i:c:s0047259x20302955
    DOI: 10.1016/j.jmva.2020.104714
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    References listed on IDEAS

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    1. Chiani, Marco, 2014. "Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy–Widom distribution," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 69-81.
    2. 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.
    3. Díaz-García, José A. & Jáimez, Ramón Gutierrez & Mardia, Kanti V., 1997. "Wishart and Pseudo-Wishart Distributions and Some Applications to Shape Theory," Journal of Multivariate Analysis, Elsevier, vol. 63(1), pages 73-87, October.
    4. Arjun K. Gupta & Daya K. Nagar, 2000. "Matrix-variate beta distribution," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 24, pages 1-11, January.
    5. 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.
    6. Bodnar, Taras & Okhrin, Yarema, 2008. "Properties of the singular, inverse and generalized inverse partitioned Wishart distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2389-2405, November.
    7. Khatri, C. G., 1972. "On the exact finite series distribution of the smallest or the largest root of matrices in three situations," Journal of Multivariate Analysis, Elsevier, vol. 2(2), pages 201-207, June.
    8. Hashiguchi, Hiroki & Nakagawa, Shigekazu & Niki, Naoto, 2000. "Simplification of the Laplace–Beltrami operator," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 51(5), pages 489-496.
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    Cited by:

    1. Shimizu, Koki & Hashiguchi, Hiroki, 2022. "Algorithm for the product of Jack polynomials and its application to the sphericity test," Statistics & Probability Letters, Elsevier, vol. 187(C).

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