On the largest eigenvalue of Wishart matrices with identity covariance when n, p and p/n tend to infinity

By Noureddine El Karoui. 2003

99 citations recorded by the source corpus.

Let X be a n*p matrix and l_1 the largest eigenvalue of the covariance matrix X^{*}*X. The "null case" where X_{i,j} are independent Normal(0,1) is of particular interest for principal component analysis. For this model, when n, p tend to infinity and n/p tends to gamma in (0,\infty), it was shown in Johnstone (2001) that l_1, properly centered and scaled, converges to the Tracy-Widom law. We show that with the same centering and scaling, the result is true even when p/n or n/p tends to infinity. The derivation uses ideas and techniques quite similar to the ones presented in Johnstone (2001). Following Soshnikov (2002), we also show that the same is true for the joint distribution of the k largest eigenvalues, where k is a fixed integer. Numerical experiments illustrate the fact that the Tracy-Widom approximation is reasonable even when one of the dimension is "small".

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/814687baf87f525446bf652795d9c62c273ddf57

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Citations in the corpus

Test for Bandedness of High Dimensional Covariance Matrices with Bandwidth Estimation (2012, unknown): “1 (2001) and El Karoui (2011).”
High Dimensional Statistical Inference and Random Matrices (2006, background): “The assumption that p/n → γ ∈ (0,∞) can be weakened to min{n, p} → ∞, as shown by El Karoui [37] – this extension is of considerable statistical importance since in many settings p≫ n (see for example [40] in these proceedings).”
Random matrix theory in statistics: A review (2014, methodology): “Paul (2011) used the technique of El Karoui (2003) to prove that under the latter setting, the scaling limits of the normalized smallest eigenvalues are reflected Tracy– Widom laws, which paralleled a result of Baker et al. (1998) that showed that the reflection of F2 about the origin is the…”
Universality results for largest eigenvalues of some sample covariance matrix ensembles (2007, methodology): “The detail of the derivation of such a result from formula (7), including the required asymptotics of correlation functions for Wishart ensembles, can be found in [29], [30] and [6].”
TESTING HYPOTHESES ABOUT THE NUMBER OF FACTORS IN LARGE FACTOR MODELS (2009, background): “El Karoui (2006) obtains such a generalization in the special spherical Wishart case.”
On the principal components of sample covariance matrices (2014, background): “The assumption that φ be fixed was relaxed in [17,35].”
Approximation of Rectangular Beta-Laguerre Ensembles and Large Deviations (2013, background): “For largest eigenvalue λmax, Johansson (2000), Johnstone (2001) and Karoui (2003) obtain its limiting distribution as β = 1, 2, 4 and γ ∈ [0,∞].”