Rare-event analysis for extremal eigenvalues of white wishart matrices

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Abstract

In this paper, we consider the extreme behavior of the extremal eigenvalues of white Wishart matrices, which plays an important role in multivariate analysis. In particular, we focus on the case when the dimension of the feature p is much larger than or comparable to the number of observations n, a common situation in modern data analysis. We provide asymptotic approximations and bounds for the tail probabilities of the extremal eigenvalues. Moreover, we construct efficient Monte Carlo simulation algorithms to compute the tail probabilities. Simulation results show that our method has the best performance among known approximation approaches, and furthermore provides an efficient and accurate way for evaluating the tail probabilities in practice.

Original languageEnglish (US)
Pages (from-to)1609-1637
Number of pages29
JournalAnnals of Statistics
Volume45
Issue number4
DOIs
StatePublished - Aug 2017

Bibliographical note

Funding Information:
Supported in part by NSF Grants DMS-12-08982 and DMS-14-06279. Supported in part by NSF Grants CMMI-1362236 and DMS-12-24362. Supported in part by an NSA grant.?%blankline%

Keywords

  • Extremal eigenvalues
  • Importance sampling
  • Random matrix
  • β-Laguerre ensemble

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