Asymptotics of empirical eigenstructure for high dimensional spiked covariance

Abstract

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.

Efficient estimation of linear functionals of principal components

“…We emphasize here that we regard a and σ 0 as fixed constants, but r, Σ and g r may all possibly depend on n. For example, this allows that Σ → ∞ as long asḡ r → ∞ at the same rate as it is the case in factor models as considered in [37]. Note that some additional conditions on r, a, σ 0 , u are needed for the class S (r) (r, a, σ 0 , u) to be nonempty.…”

FarmTest: Factor-Adjusted Robust Multiple Testing With Approximate False Discovery Control

“…The proof, which is given in in Appendix D, is based on Weyl's inequality and a useful variant of the Davis-Kahan theorem . We notice that some preceding works (Onatski, 2012;Shen et al, 2016;Wang and Fan, 2017) have provided similar results under a weaker pervasiveness assumption which allows p/n → ∞ in any manner and the spiked eigenvalues {λ } K =1 are allowed to grow slower than p so long as c = p/(nλ ) is bounded.…”

Wald Statistics in high-dimensional PCA

“…Subsequent work in this model has mainly been focused on the failures of PCA in high-dimensions when the dimension p → ∞ and p/n → const. [7,15,13,21]. A remedy is to assume that the leading eigenvectors θ i in (1.1) are sparse, enabling thus inference even when p/n → ∞ [3,5,18,2,20].…”