Order determination for large dimensional matrices and its application

With the technical advancement, the collected data often involve many variables, which consists of a high-dimensional vector. Reducing the dimensionality for further data analysis is often in demand. In this project, we investigate the order determination for large dimensional matrices when the sample size is proportional to the number of variables. The examples include spiked population models, spiked Fisher models, principal component analysis and canonical correlation analysis. Unlike the models with xed number of variables, the asymptotic properties of the matrices behave very dierent and the order determination becomes dicult. Thus, we will propose some new methods to determine the number of spikes, the number of principal components, the number of factors in approximate factor models or the number of pairs of canonical variable and give a systematic study to show when the orders can be consistently estimated and when cannot. Numerical studies will be conducted to examine the nite sample performance of the methods. 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 elds.