Thresholding double ridge ratio criteria for order determination and their applications

Popularly used eigendecomposition-based criteria such as BIC type, ratio estimation and principal component-based criterion often underdetermine model order for semiparametric regression models, multivariate linear models, approximate factor models, principal component analysis and independent component analysis. This longstanding problem is mainly caused by the existence of one or two dominating eigenvalues compared to other nonzero eigenvalues.

To alleviate this difficulty, we propose thresholding double ridge ratio criteria such that the true orders in different problems can be better identified and are less underdetermined. Unlike all existing eigendecomposition-based criteria, these types of criteria could define consistent estimates without requiring the existence of unique minimum or maximum and can then handle possible multiple local minima or maxima scenarios. This generic strategy would be readily applied to the problems with functional data or tensor data. In this project, we investigate the order determination for general sufficient dimension reduction theory with fixed and divergent dimensions; for local models that converge to its limiting model with fewer projected covariates, discuss when the order can be consistently estimated, when cannot; and for approximate factor models, study the estimation consistency for the number of common factors. The method will also be applied to functional data and tensor data.

Numerical studies will be conducted to examine the finite sample performance of the method. This research project investigates an important but longstanding problem in order determination in several models, and expect the new methodology and theory developed in the course of the project will have a lasting impact in the relevant fields.