Asymptotics of Empirical Eigen-structure for Ultra-high Dimensional Spiked Covariance Model

TL;DR

This paper develops a comprehensive asymptotic theory for the eigenvalues and eigenvectors of high-dimensional spiked covariance matrices, revealing biases and proposing a new estimator for improved PCA and risk assessment.

Contribution

It introduces a unified asymptotic regime for high-dimensional eigen-structure analysis, extending prior work and addressing convergence rates and bias correction in PCA.

Findings

Derived asymptotic distributions for eigenvalues and eigenvectors.

Proposed the S-POET estimator for bias correction.

Validated results through simulations and applications.

Abstract

We derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime, which takes into account the spike magnitude of leading eigenvalues, sample size, and dimensionality. This new regime allows high dimensionality and diverging eigenvalue spikes and provides new insights into the roles the leading eigenvalues, sample size, and dimensionality play in principal component analysis. The results are proven by a technical device, which swaps the role of rows and columns and converts the high-dimensional problems into low-dimensional ones. Our results are a natural extension of those in Paul (2007) to more general setting with new insights and solve the rates of convergence problems in Shen et al. (2013). They also reveal the biases of the estimation of 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.