Accuracy of the Tracy-Widom limits for the extreme eigenvalues in white Wishart matrices
نویسنده
چکیده
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 Wp(I, n), we study the accuracy of their scaling limits under the setting: n/p → γ ∈ (0,∞) as n → ∞. 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 O(min(n, p)−2/3). If γ > 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% percent in ‘thin’ data matrix settings, even with min(n, p) as small as 2.
منابع مشابه
Accuracy of the Tracy-Widom limit for the largest eigenvalue in white Wishart matrices
Let A be a p-variate real Wishart matrix on n degrees of freedom with identity covariance. The distribution of the largest eigenvalue in A has important applications in multivariate statistics. Consider the asymptotics when p grows in proportion to n, it is known from Johnstone (2001) that after centering and scaling, these distributions approach the orthogonal Tracy-Widom law for real-valued d...
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