# Central limit theorems for eigenvalues in a spiked population model

@article{Bai2008CentralLT, title={Central limit theorems for eigenvalues in a spiked population model}, author={Zhidong Bai and Jian-Feng Yao}, journal={Annales De L Institut Henri Poincare-probabilites Et Statistiques}, year={2008}, volume={44}, pages={447-474} }

In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). This model is proposed by Johnstone to cope with empirical findings on various data sets. The question is to quantify the effect of the perturbation caused by the spike eigenvalues. A recent work by Baik and Silverstein establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the…

## 188 Citations

On determining the number of spikes in a high-dimensional spiked population model

- Mathematics
- 2011

In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). Determining the number of spikes is a fundamental…

Title On sample eigenvalues in a generalized spiked populationmodel

- Mathematics
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In the spiked population model introduced by Johnstone [11], the population covariance matrix has all its eigenvalues equal to unit except for a few fixed eigenvalues (spikes). The question is to…

On sample eigenvalues in a generalized spiked population model

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- 2012

The limits of the sample spiked eigenvalues for a high-dimensional generalized Fisher matrix and its applications.

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- 2019

Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices

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We study the asymptotic distributions of the spiked eigenvalues and the largest nonspiked eigenvalue of the sample covariance matrix under a general covariance matrix model with divergent spiked…

The largest eigenvalues of sample covariance matrices for a spiked population: Diagonal case

- Mathematics
- 2009

We consider large complex random sample covariance matrices obtained from “spiked populations,” that is, when the true covariance matrix is diagonal with all but finitely many eigenvalues equal to 1.…

Asymptotic independence of spiked eigenvalues and linear spectral statistics for large sample covariance matrices

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- 2020

We consider general high-dimensional spiked sample covariance models and show that their leading sample spiked eigenvalues and their linear spectral statistics are asymptotically independent when the…

ASYMPTOTIC INDEPENDENCE OF SPIKED EIGENVALUES AND LINEAR SPECTRAL STATISTICS FOR LARGE SAMPLE COVARIANCE MATRICES

- Mathematics
- 2022

We consider general high-dimensional spiked sample covariance models and show that their leading sample spiked eigenvalues and their linear spectral statistics are asymptotically independent when the…

INVARIANCE PRINCIPLE AND CLT FOR THE SPIKED EIGENVALUES OF LARGE-DIMENSIONAL FISHER MATRICES AND APPLICATIONS

- Mathematics
- 2022

This paper aims to derive asymptotical distributions of the spiked eigenvalues of the large-dimensional spiked Fisher matrices without Gaussian assumption and the restrictive assumptions on…

A note on the CLT of the LSS for sample covariance matrix from a spiked population model

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- 2014

## References

SHOWING 1-10 OF 16 REFERENCES

Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices

- Mathematics
- 2004

AbstractWe compute the limiting distributions of the largest eigenvalue of a complex Gaussian samplecovariance matrix when both the number of samples and the number of variables in each samplebecome…

No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices

- Mathematics
- 1998

Let B n = (1/N)T n 1/2 X n X n *Tn 1/2 , where X n is n x N with i.i.d. complex standardized entries having finite fourth moment and T n 1/2 is a Hermitian square root of the nonnegative definite…

METHODOLOGIES IN SPECTRAL ANALYSIS OF LARGE DIMENSIONAL RANDOM MATRICES , A REVIEW

- Mathematics
- 1999

In this paper, we give a brief review of the theory of spectral analysis of large dimensional random matrices. Most of the existing work in the literature has been stated for real matrices but the…

CLT FOR LINEAR SPECTRAL STATISTICS OF LARGE-DIMENSIONAL SAMPLE COVARIANCE MATRICES

- Mathematics
- 2008

Let Bn = (1/N)T 1/2 n XnX∗ nT 1/2 n where Xn = (Xij ) is n × N with i.i.d. complex standardized entries having finite fourth moment, and T 1/2 n is a Hermitian square root of the nonnegative definite…

DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES

- Mathematics
- 1967

In this paper we study the distribution of eigenvalues for two sets of random Hermitian matrices and one set of random unitary matrices. The statement of the problem as well as its method of…

On the distribution of the largest eigenvalue in principal components analysis

- Mathematics
- 2001

Let x (1) denote the square of the largest singular value of an n x p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x (1) is the largest principal component…

A reliable data-based bandwidth selection method for kernel density estimation

- Computer Science
- 1991

The key to the success of the current procedure is the reintroduction of a non- stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance.

Matrix analysis

- MathematicsStatistical Inference for Engineers and Data Scientists
- 2018

This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications.

Random Matrices

- Computer Science
- 2005

This workshop was unusually diverse, even by MSRI standards; the attendees included analysts, physicists, number theorists, probabilists, combinatorialists, and more.