Accuracy of the Tracy--Widom limits for the extreme eigenvalues in white Wishart matrices

Details Accuracy of the Tracy--Widom limits for the extreme eigenvalues in white Wishart matrices Accuracy of the Tracy--Widom limits for the extreme eigenvalues in white Wishart matrices

2012-03-05

arxiv.org/abs/1203.0839v1

Bernoulli 2012, Vol. 18, No. 1, 322-359

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/10-BEJ334 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/10-BEJ334

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 $W_p(I,n)$, we study the accuracy of their scaling limits under the setting: $n/p\rightarrow \gamma\in(0,\infty)$ as $n\rightarrow \infty$. 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 ${\mathrm {O}({\min(n,p)^{-2/3}})}$. If $\gamma>1$, 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.

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