Accuracy of the TracyWidom limits for the extreme eigenvalues in white Wishart matrices
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 $W_p(I,n)$, 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 TracyWidom 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2012
 DOI:
 10.48550/arXiv.1203.0839
 arXiv:
 arXiv:1203.0839
 Bibcode:
 2012arXiv1203.0839M
 Keywords:

 Mathematics  Statistics Theory
 EPrint:
 Published in at http://dx.doi.org/10.3150/10BEJ334 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)