Statistical inference for principal components of spiked covariance matrices

In this paper, we study the asymptotic behavior of extreme eigenvalues and eigenvectors high-dimensional spiked sample covariance matrices, in supercritical case when a reliable detection spikes is possible. particular, derive joint distribution generalized components associated eigenvectors, that is, projections onto arbitrary given direction, assuming dimension size are comparably large. general, terms linear combinations finitely many Gaussian Chi-square variables, with parameters depending on projection direction spikes. Our assumption fully general. First, strengths only required to be slightly above critical threshold no upper bound needed. Second, multiple spikes, same strength, allowed. Third, structural imposed Thanks general setting, can then apply results various high dimensional statistical hypothesis testing problems involving both eigenvectors. Specifically, propose accurate powerful statistics conduct principal components. These data-dependent adaptive underlying true Numerical simulations also confirm accuracy powerfulness our proposed illustrate significantly better performance compared existing methods literature. even either small or