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

We derived the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime, which takes into account the spikeness of leading eigenvalues, sample size, and dimensionality. This new regime allows high dimensionality and diverging eigenvalue spikes and provides new insights on the roles the leading eigenvalues, sample size, and dimensionality played in the principal component analysis. The results are proven by a new 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 the principal component analysis, and lead to a new covariance estimator for the approximate factor model, called shrinkage principal orthogonal complement thresholding (S-POET), which 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.