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Estimating tail probabilities of the ratio of the largest eigenvalue to the trace of a Wishart matrix

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  • He, Yinqiu
  • Xu, Gongjun

Abstract

This paper develops an efficient Monte Carlo method to estimate the tail probabilities of the ratio of the largest eigenvalue to the trace of the Wishart matrix, which plays an important role in multivariate data analysis. The estimator is constructed based on a change-of-measure technique and it is proved to be asymptotically efficient for both the real and complex Wishart matrices. Simulation studies further show the improved performance of the proposed method over existing approaches based on asymptotic approximations, especially when estimating probabilities of rare events.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:166:y:2018:i:c:p:320-334
    DOI: 10.1016/j.jmva.2018.03.011
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    References listed on IDEAS

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    1. Davis, A. W., 1972. "On the ratios of the individual latent roots to the trace of a Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 2(4), pages 440-443, December.
    2. Schuurmann, F. J. & Krishnaiah, P. R. & Chattopadhyay, A. K., 1973. "On the distributions of the ratios of the extreme roots to the trace of the Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 3(4), pages 445-453, December.
    3. 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.
    4. 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. Yangqing Deng & Yinqiu He & Gongjun Xu & Wei Pan, 2022. "Speeding up Monte Carlo simulations for the adaptive sum of powered score test with importance sampling," Biometrics, The International Biometric Society, vol. 78(1), pages 261-273, March.

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