Annals of Statistics

Rare-event analysis for extremal eigenvalues of white Wishart matrices

Tiefeng Jiang, Kevin Leder, and Gongjun Xu

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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.

Article information

Source
Ann. Statist., Volume 45, Number 4 (2017), 1609-1637.

Dates
Received: April 2015
Revised: July 2016
First available in Project Euclid: 28 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.aos/1498636868

Digital Object Identifier
doi:10.1214/16-AOS1502

Mathematical Reviews number (MathSciNet)
MR3670190

Zentralblatt MATH identifier
1377.65013

Subjects
Primary: 65C05: Monte Carlo methods 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Keywords
Importance sampling extremal eigenvalues random matrix $\beta$-Laguerre ensemble

Citation

Jiang, Tiefeng; Leder, Kevin; Xu, Gongjun. Rare-event analysis for extremal eigenvalues of white Wishart matrices. Ann. Statist. 45 (2017), no. 4, 1609--1637. doi:10.1214/16-AOS1502. https://projecteuclid.org/euclid.aos/1498636868


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Supplemental materials

  • Supplement to “Rare-event analysis for extremal eigenvalues of white Wishart matrices.”. The online Supplementary Material contains proofs of technical lemmas (Lemmas 1–9) and Theorem 3.