Abstract: Motivated by dimension reduction in the context of regression analysis and signal detection, we investigate the order determination for large-dimensional matrices, including spiked-type models, in which the numbers of covariates are proportional to the sample sizes for different models. Because the asymptotic behaviors of the estimated eigenvalues of the corresponding matrices differ from those in fixed-dimension scenarios, we discuss the largest possible number we can identify and introduce a "valley-cliff" criterion. We propose two versions of the criterion. The first is based on the original differences between the eigenvalues. The second is based on the transformed differences between the eigenvalues, which reduces the effect of the ridge selection in the former case. This generic method is very easy to implement and computationally inexpensive, and can be applied to various matrices. As examples, we focus on spiked population models, spiked Fisher matrices, and factor models with auto-covariance matrices. Numerical studies are conducted to examine the finite-sample performance of the method, which we compare with that of existing methods.

Key words and phrases: Auto-covariance matrix, factor model, finite-rank perturbation, Fisher matrix, phase transition, ridge ratio, spiked population model.