Open Access
February 2012 Accuracy of the Tracy–Widom limits for the extreme eigenvalues in white Wishart matrices
Zongming Ma
Bernoulli 18(1): 322-359 (February 2012). DOI: 10.3150/10-BEJ334

Abstract

The distributions of the largest and the smallest eigenvalues of a p-variate sample covariance matrix S are of great importance in statistics. Focusing on the null case where nS follows the standard Wishart distribution Wp(I, n), we study the accuracy of their scaling limits under the setting: n/pγ ∈ (0, ∞) as n → ∞. The limits here are the orthogonal Tracy–Widom law and its reflection about the origin.

With carefully chosen rescaling constants, the approximation to the rescaled largest eigenvalue distribution by the limit attains accuracy of order O(min(n, p)−2/3). If γ > 1, the same order of accuracy is obtained for the smallest eigenvalue after incorporating an additional log transform. Numerical results show that the relative error of approximation at conventional significance levels is reduced by over 50% in rectangular and over 75% in ‘thin’ data matrix settings, even with min(n, p) as small as 2.

Citation

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Zongming Ma. "Accuracy of the Tracy–Widom limits for the extreme eigenvalues in white Wishart matrices." Bernoulli 18 (1) 322 - 359, February 2012. https://doi.org/10.3150/10-BEJ334

Information

Published: February 2012
First available in Project Euclid: 20 January 2012

zbMATH: 1248.60010
MathSciNet: MR2888709
Digital Object Identifier: 10.3150/10-BEJ334

Keywords: Eigenvalues of random matrices , Laguerre orthogonal ensemble , Principal Component Analysis , rate of convergence , Tracy–Widom distribution , Wishart distribution

Rights: Copyright © 2012 Bernoulli Society for Mathematical Statistics and Probability

Vol.18 • No. 1 • February 2012
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