CONSISTENCY OF AIC AND BIC IN ESTIMATING THE NUMBER OF SIGNIFICANT COMPONENTS IN HIGH-DIMENSIONAL PRINCIPAL COMPONENT ANALYSIS

• 成果类型: Article
• 署名作者: Bai, Zhidong; Choi, Kwok Pui; Fujikoshi, Yasunori
• 署名单位: Northeast Normal University - China; Northeast Normal University - China; National University of Singapore; Hiroshima University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1577
• 发表日期: 2018
• 页码: 1050-1076
• 关键词: smallest eigenvalues; COVARIANCE-MATRIX; model selection; linear-model; criteria; regression; EQUALITY

摘要

In this paper, we study the problem of estimating the number of significant components in principal component analysis (PCA), which corresponds to the number of dominant eigenvalues of the covariance matrix of p variables. Our purpose is to examine the consistency of the estimation criteria AIC and BIC based on the model selection criteria by Akaike [In 2nd International Symposium on Information Theory (1973) 267-281, Akademia Kiado] and Schwarz [Estimating the dimension of a model 6 (1978) 461464] under a high-dimensional asymptotic framework. Using random matrix theory techniques, we derive sufficient conditions for the criterion to be strongly consistent for the case when the dominant population eigenvalues are bounded, and when the dominant eigenvalues tend to infinity. Moreover, the asymptotic results are obtained without normality assumption on the population distribution. Simulation studies are also conducted, and results show that the sufficient conditions in our theorems are essential.