Limiting laws and consistent estimation criteria for fixed and diverging number of spiked eigenvalues

TL;DR

This paper develops new limiting laws and estimation criteria for the number of spiked eigenvalues in high-dimensional covariance matrices, applicable under weaker conditions and without normality assumptions.

Contribution

It introduces a generalized estimation criterion that remains consistent for both fixed and diverging numbers of spiked eigenvalues, improving upon existing methods.

Findings

Consistent estimation of the number of spikes under weaker conditions.

Derived limiting distributions without boundedness assumptions.

Estimation criteria effective for diverging and fixed spike counts.

Abstract

In this paper, we study limiting laws and consistent estimation criteria for the extreme eigenvalues in a spiked covariance model of dimension $p$. Firstly, for fixed $p$, we propose a generalized estimation criterion that can consistently estimate, $k$, the number of spiked eigenvalues. Compared with the existing literature, we show that consistency can be achieved under weaker conditions on the penalty term. Next, allowing both $p$ and $k$ to diverge, we derive limiting distributions of the spiked sample eigenvalues using random matrix theory techniques. Notably, our results do not require the spiked eigenvalues to be uniformly bounded from above or tending to infinity, as have been assumed in the existing literature. Based on the above derived results, we formulate a generalized estimation criterion and show that it can consistently estimate $k$, while $k$ can be fixed or grow at an order of $k=o(n^{1/3})$. We further show that the results in our work continue to hold under a general population distribution without assuming normality. The efficacy of the proposed estimation criteria is illustrated through comparative simulation studies.