Accuracy of the Tracy--Widom 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: $n/p\to \gamma \in (0,\infty )$ as $n\to \infty$. 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.

arXiv commentPublished in at http://dx.doi.org/10.3150/10-BEJ334 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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• Fast approach to the Tracy-Widom law at the edge of GOE and GUE
• On the probability that all eigenvalues of Gaussian, Wishart, and double Wishart random matrices lie within an interval
• Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices
• Limits on Sparse Data Acquisition: RIC Analysis of Finite Gaussian Matrices
• Recursion for the smallest eigenvalue density of $\beta$-Wishart-Laguerre ensemble
• Optimal soft edge scaling variables for the Gaussian and Laguerre even $\beta$ ensembles
• Limiting distribution of maximal crossing and nesting of Poissonized random matchings
• Functional form for the leading correction to the distribution of the largest eigenvalue in the GUE and LUE
• Convergence rate to the Tracy-Widom laws for the largest eigenvalue of Wigner matrices
• Asymptotic Expansions of Gaussian and Laguerre Ensembles at the Soft Edge II: Level Densities