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

作者：Ma Zongming^*

来源：Bernoulli, 2012, 18(1): 322-359.

DOI：10.3150/10-BEJ334

• eigenvalues of random matrices
• Laguerre orthogonal ensemble
• principal component analysis
• rate of convergence
• Tracy-Widom distribution
• Wishart distribution

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 -> gamma epsilon (0, infinity) as n -> infinity. The limits here are the orthogonal Tracy-Widom law and its reflection about the origin.
With carefully chosen resealing constants, the approximation to the resealed largest eigenvalue distribution by the limit attains accuracy of order 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.

• 出版日期2012-2