Central limit theorems for eigenvalues in a spiked population model

By Z. Bai, Jianfeng Yao. 2008

238 citations recorded by the source corpus.

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 sample sizes become large. This paper establishes the limiting distributions of these extreme sample eigenvalues. As another important result of the paper, we provide a central limit theorem on random sesquilinear forms.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/dffe01ea16dc103974b64f391d09e80d447dc02b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

Identifying the number of factors from singular values of a large sample auto-covariance matrix (2014, methodology): “For some recent accounts on this theory, we refer to Johnstone [15], Baik and Silverstein [6], Bai and Yao [4], Benaych-Georges and Nadakuditi [7], Passemier and Yao [19] and the references therein.”
Almost sure localization of the eigenvalues in a Gaussian information plus noise model - Application to the spiked models (2010, background): “Bai and Yao showed in [1] that certain eigenvalues of a non Gaussian spiked model satisfy a central limit theorem.”
Singular vector and singular subspace distribution for the matrix denoising model (2018, background): “The fluctuation of the extreme eigenvalues of various models have been considered in [2, 3, 7, 8, 9, 23, 24, 28, 14, 15, 42, 55, 34, 56, 59, 44].”
On finite rank deformations of Wigner matrices (2011, background): “We also note several papers on the eigenvalues of sample covariance matrices of spiked population models ([3,5,6,26]).”
Limits of spiked random matrices I (2010, background): “tigations into the limiting kernels (Desrosiers and Forrester [19]), and generalizations beyond the spiked model (El Karoui [23]) and away from Gaussianity (Bai and Yao [7], Féral and Péché [26]) (Telatar [52]).”
On sample eigenvalues in a generalized spiked population model (2008, methodology): “In a recent work Bai and Yao [5], considering general matrix entries as in [6], we have established central limit theorems for these extreme sam- ple eigenvalues generated by spike eigenvalues which are outside the critical interval [1 −√y, 1 + √y].”
Large deviations of the extreme eigenvalues of random deformations of matrices (2010, background): “(2) If G is such that for any α > 0, E[eα ∑r i=1 |gi|2] ∞, then the second point of Assumption 2.6 holds with I infinite except at 0, where it takes value 0.”
On the principal components of sample covariance matrices (2014, methodology): “2 with fixed nonzero {di} and φ was investigated in [2, 4].”
Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices (2017, background): “Bai and Yao [8] further showed that the spiked eigenvalues converge in distribution to Gaussian distribution or the eigenvalues of a finite dimensional”
On estimation of the noise variance in high dimensional probabilistic principal component analysis (2013, background): “ low-dimensional limits n;k ! i+ ˙2 (population spike eigenvalues) and n;k!˙2 (population noise eigenvalues) discussed earlier. In a further step, CLT for the spiked eigenvalues is established in Bai and Yao (2008) (see also Paul (2007)): the n i-dimensional vector f p n( n;k ˙2˚( k )),k2J ig (8) ”
The singular values and vectors of low rank perturbations of large rectangular random matrices (2011, background): “1 of [9]), where no hypothesis of symmetry or about the third moment is made.”
The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations. (2007, background): “This result is in fact a particular case of a more general result of [6] (Theorems 7.”
Outliers in the spectrum of iid matrices with bounded rank perturbations (2010, background): “ X n+ C n) z) Q n j=1 ( j( p1 n X n) z) whenever the denominator is nonzero. We are indebted to Alice Guionnet for suggesting the use of this type of criterion, versions of which also appear in [3], [8], [18], [13], [14], [15]. For instance, the k= 1 case of this criterion appears explicitly in [13]”
Universality for the largest eigenvalue of sample covariance matrices with general population (2013, background): “For generally distributed spiked sample covariance matrices, the universality property was also partially established in [2] and [21].”