Analysis Of Eigenvalues In Sample Covariance Matrices Of Dynamic Spiked Population Models

The spiked population model, proposed by Johnstone, is a useful model in terms of dealing with high-dimensional data. In a spiked population model, all the population eigenvalues are one except for a fix eigenvalues which are called spikes. Some extreme eigenvalues of sample covariance matrices of spiked population models, in the high-dimensional setting, would converge almost surely and have limiting distribution. This paper introduces a new model named dynamic spiked population model by removing the restriction of the number of spikes in the previous model. Then, we propose almost surely convergence of these extreme eigenvalues of sample covariance matrices for this new model. The question is how to estimate the number of spikes via information of sample covariance matrices. Passemier and Yao proposed methods of estimating for spiked population model and proved the consistency of these estimator. Considering differences of estimation between these two models, this paper presents a new estimator, which is based on the almost surely convergence of extreme sample eigenvalues. This estimator has a simple form and performs well in simulation experiments.