Asymptotics of empirical eigenstructure for high dimensional spiked covariance

Weichen Wang Jianqing Fan

Statistics Theory and Methods mathscidoc:1912.43324

Annals of statistics, 45, (3), 1342, 2017.6
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
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  title={Asymptotics of empirical eigenstructure for high dimensional spiked covariance},
  author={Weichen Wang, and Jianqing Fan},
  booktitle={Annals of statistics},
Weichen Wang, and Jianqing Fan. Asymptotics of empirical eigenstructure for high dimensional spiked covariance. 2017. Vol. 45. In Annals of statistics. pp.1342.
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