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Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy–Widom distribution

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  • Chiani, Marco

Abstract

We derive efficient recursive formulas giving the exact distribution of the largest eigenvalue for finite dimensional real Wishart matrices and for the Gaussian Orthogonal Ensemble (GOE). In comparing the exact distribution with the limiting distribution of large random matrices, we also found that the Tracy–Widom law can be approximated by a properly scaled and shifted gamma distribution, with great accuracy for the values of common interest in statistical applications.

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  • 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.
    • Handle: RePEc:eee:jmvana:v:129:y:2014:i:c:p:69-81
    DOI: 10.1016/j.jmva.2014.04.002
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    References listed on IDEAS

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    1. C. Khatri, 1969. "Non-central distributions ofith largest characteristic roots of three matrices concerning complex multivariate normal populations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 21(1), pages 23-32, December.
    2. Nadler, Boaz, 2011. "On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 363-371, February.
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    Cited by:

    1. Dharmawansa, Prathapasinghe & Nadler, Boaz & Shwartz, Ofer, 2019. "Roy’s largest root under rank-one perturbations: The complex valued case and applications," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    2. He, Yinqiu & Xu, Gongjun, 2018. "Estimating tail probabilities of the ratio of the largest eigenvalue to the trace of a Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 320-334.
    3. Azaïs, Jean-Marc & Delmas, Céline, 2022. "Mean number and correlation function of critical points of isotropic Gaussian fields and some results on GOE random matrices," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 411-445.
    4. Chiani, Marco, 2016. "Distribution of the largest root of a matrix for Roy’s test in multivariate analysis of variance," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 467-471.
    5. 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).

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