PCA consistency for the power spiked model in high-dimensional settings

Yata, Kazuyoshi; Aoshima, Makoto

doi:10.1016/j.jmva.2013.08.003

Cited by 35 publications

(42 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the one hand, they help quantify the biases of empirical eigen-structure and explain where they come from. Specifically, in Theorem 3.1, the bias of the j th sample eigenvalue ( j ≤ m ) is quantified by p /( n λ j ), which is also showed by Yata and Aoshima (2012, 2013) under different assumptions of the spiked covariance model. Our novel contribution lies in Theorem 3.2, revealing the bias of the j th sample eigenvector ( j ≤ m ).…”

Section: Introductionmentioning

confidence: 85%

See 1 more Smart Citation

Asymptotics of empirical eigenstructure for high dimensional spiked covariance

2017

View full text Add to dashboard Cite

We derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime, which takes into account the magnitude of spiked eigenvalues, sample size, and dimensionality. This regime allows high dimensionality and diverging eigenvalues and provides new insights into the roles that the leading eigenvalues, sample size, and dimensionality play in principal component analysis. Our results are a natural extension of those in Paul (2007) to a more general setting and solve the rates of convergence problems in Shen et al. (2013). They also reveal the biases of estimating leading eigenvalues and eigenvectors by using principal component analysis, and lead to a new covariance estimator for the approximate factor model, called shrinkage principal orthogonal complement thresholding (S-POET), that corrects the biases. Our results are successfully applied to outstanding problems in estimation of risks of large portfolios and false discovery proportions for dependent test statistics and are illustrated by simulation studies.

show abstract

Section: Introductionmentioning

confidence: 85%

“…Our result here holds for each individual spike. Yata and Aoshima (2012, 2013) employed a similar technical trick and gave a comprehensive study on the asymptotic consistency and distributions of the eigenvalues. They got similar results under different conditions from ours.…”

Section: Asymptotic Behavior Of Empirical Eigen-structurementioning

confidence: 99%

Asymptotics of empirical eigenstructure for high dimensional spiked covariance

2017

View full text Add to dashboard Cite

show abstract

“…Deeper related results, under various assumptions including conditions only on the covariance matrix, are available in the above referenced series of papers by Yata and Aoshima (2009, 2010a, 2012a, 2013). Related new insights about sparse PCA were dicovered by Shen et al (2012a).…”

Section: Pca and Hdlss Asymptoticsmentioning

confidence: 99%

“…The case where n grows more slowly than d has been called High Dimension Moderate Sample Size by Borysov et al (2014). Yata and Aoshima (2010a, 2012a, 2013) have also invented modified versions of PCA which provide eigenvalue estimates that are consistent.…”

Section: Pca and Hdlss Asymptoticsmentioning

confidence: 99%

See 1 more Smart Citation

The statistics and mathematics of high dimension low sample size asymptotics

Shen¹,

Shen²,

Zhu³

et al. 2017

STAT SINICA

View full text Add to dashboard Cite

The aim of this paper is to establish several deep theoretical properties of principal component analysis for multiple-component spike covariance models. Our new results reveal an asymptotic conical structure in critical sample eigendirections under the spike models with distinguishable (or indistinguishable) eigenvalues, when the sample size and/or the number of variables (or dimension) tend to infinity. The consistency of the sample eigenvectors relative to their population counterparts is determined by the ratio between the dimension and the product of the sample size with the spike size. When this ratio converges to a nonzero constant, the sample eigenvector converges to a cone, with a certain angle to its corresponding population eigenvector. In the High Dimension, Low Sample Size case, the angle between the sample eigenvector and its population counterpart converges to a limiting distribution. Several generalizations of the multi-spike covariance models are also explored, and additional theoretical results are presented.

show abstract