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June 2018 Consistency of AIC and BIC in estimating the number of significant components in high-dimensional principal component analysis
Zhidong Bai, Kwok Pui Choi, Yasunori Fujikoshi
Ann. Statist. 46(3): 1050-1076 (June 2018). DOI: 10.1214/17-AOS1577

Abstract

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, Akadémia Kiado] and Schwarz [Estimating the dimension of a model 6 (1978) 461–464] 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.

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Zhidong Bai. Kwok Pui Choi. Yasunori Fujikoshi. "Consistency of AIC and BIC in estimating the number of significant components in high-dimensional principal component analysis." Ann. Statist. 46 (3) 1050 - 1076, June 2018. https://doi.org/10.1214/17-AOS1577

Information

Received: 1 October 2015; Revised: 1 January 2017; Published: June 2018
First available in Project Euclid: 3 May 2018

zbMATH: 1395.62119
MathSciNet: MR3797996
Digital Object Identifier: 10.1214/17-AOS1577

Subjects:
Primary: 62H12
Secondary: 62H30

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 3 • June 2018
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