On the border of extreme and mild spiked models in the HDLSS framework

Abstract

In the spiked covariance model for High Dimension Low Sample Size (HDLSS) asymptotics where the dimension tends to infinity while the sample size is fixed, a few largest eigenvalues are assumed to grow as the dimension increases. The rate of growth is crucial as the asymptotic behavior of the sample Principal Component (PC) directions changes dramatically, from consistency to strong inconsistency at the boundary of the extreme and mild spiked covariance models. Yet, the behavior at the boundary spiked model is unexplored. We study the HDLSS asymptotic behavior of the eigenvalues and the eigenvectors of the sample covariance matrix at the boundary spiked model and observe that they show intermediate behavior between the extreme and mild spiked models.