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

By Zongming Ma. 2012

85 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7ed584d9ff56b0940822860e9c69e4d544510d70

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In Applied Mathematics

In Statistics and Probability

Citations in the corpus

Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation (2016, methodology): “, Parzen (1957) and Samarov (1977). Another parameter space, which directly specifies the decay rate of the autocovariance sequence (σm), has also been considered in the literature, see, Cai, Ren, and Zhou (2013).”
Asymptotics and Concentration Bounds for Bilinear Forms of Spectral Projectors of Sample Covariance (2014, unknown): “The accuracy of this approximation was studied in [14, 21].”
Approximation of Rectangular Beta-Laguerre Ensembles and Large Deviations (2013, result): “Ma [45] obtains a similar result for β = 1.”
Random matrix theory in statistics: A review (2014, methodology): “…the rate of convergence of the (appropriately normalized) largest eigenvalue of the JOE and JUE to the corresponding Tracy– Widom limits is Oðn 2=3Þ. Johnstone and Ma (2012) used similar techniques to derive the Oðn 2=3Þ bound on the rate of convergence for the largest eigenvalue of GOE and GUE.”
Maximum likelihood estimation for linear Gaussian covariance models (2014, methodology): “To analyse the probability that Δ2Sn contains the true covariance matrix Σ Å, we build on the recent work of Ma (2012) who obtained improved approximations for the extreme eigenvalues of the white Wishart distribution in terms of the quantities μn,p ={.n− 12 /1=2 − .p− 12 /1=2}2, σn,p ={.n− 12 /1=2…”