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On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

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  • Nadler, Boaz

Abstract

The ratio of the largest eigenvalue divided by the trace of a p×p random Wishart matrix with n degrees of freedom and an identity covariance matrix plays an important role in various hypothesis testing problems, both in statistics and in signal processing. In this paper we derive an approximate explicit expression for the distribution of this ratio, by considering the joint limit as both p,n-->[infinity] with p/n-->c. Our analysis reveals that even though asymptotically in this limit the ratio follows a Tracy-Widom (TW) distribution, one of the leading error terms depends on the second derivative of the TW distribution, and is non-negligible for practical values of p, in particular for determining tail probabilities. We thus propose to explicitly include this term in the approximate distribution for the ratio. We illustrate empirically using simulations that adding this term to the TW distribution yields a quite accurate expression to the empirical distribution of the ratio, even for small values of p,n.

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  • 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.
    • Handle: RePEc:eee:jmvana:v:102:y:2011:i:2:p:363-371
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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.
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    Cited by:

    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. Deo, Rohit S., 2016. "On the Tracy–Widom approximation of studentized extreme eigenvalues of Wishart matrices," Journal of Multivariate Analysis, Elsevier, vol. 147(C), pages 265-272.
    3. 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.
    4. Hachem, Walid & Loubaton, Philippe & Mestre, Xavier & Najim, Jamal & Vallet, Pascal, 2013. "A subspace estimator for fixed rank perturbations of large random matrices," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 427-447.

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