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Distribution of the ratio of two Wishart matrices and cumulative probability evaluation by the holonomic gradient method

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

Abstract

We study the distribution of the ratio of two central Wishart matrices with different covariance matrices. We first derive the density function of a particular matrix form of the ratio and show that its cumulative distribution function can be expressed in terms of the hypergeometric function 2F1 of a matrix argument. Then we apply the holonomic gradient method for numerical evaluation of the hypergeometric function. This approach enables us to compute the power function of Roy’s maximum root test for testing the equality of two covariance matrices.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:165:y:2018:i:c:p:270-278
    DOI: 10.1016/j.jmva.2018.01.002
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    References listed on IDEAS

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    1. Venables, W., 1973. "Computation of the null distribution of the largest or smallest latent roots of a beta matrix," Journal of Multivariate Analysis, Elsevier, vol. 3(1), pages 125-131, March.
    2. 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.
    3. Yasuko Chikuse, 1977. "Asymptotic expansions for the joint and marginal distributions of the latent roots ofS 1 S 2 −1," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 29(1), pages 221-233, December.
    4. 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.
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    Cited by:

    1. 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).
    2. Yong Bao & Aman Ullah, 2021. "Analytical Finite Sample Econometrics: From A. L. Nagar to Now," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 19(1), pages 17-37, December.

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