Order determination for spiked-type models with a divergent number of spikes

Abstract

For large dimensional spiked models, the order (number of spikes) determination is an important issue for dimension reduction. The authors propose a generic criterion to estimate the order when the dimension is proportional to the sample size and the order is divergent as the dimension goes to infinity. To handle the divergence of the order, the criterion is defined by location-shift truncated eigenvalues, unlike the existing criteria. They suggest two versions of the criterion: the first defines an objective function that is a sequence of ridge ratios of the defined eigenvalues in order to have a clear separation between the ratio at the true order and other ratios; and the second uses an objective function of double ridge ratios to enhance such a separation. To alleviate the effect of the bias in the scale estimation when the order is large, an iterative procedure is utilized for the estimation. Numerical studies are conducted on spiked population models and spiked Fisher matrices to examine the finite sample performances of the proposed methods.