On estimation of the noise variance in high dimensional probabilistic principal component analysis. (31st December 2015)

• Record Type: Journal Article
• Authors: Passemier, Damien; Li, Zhaoyuan; Yao, Jianfeng
• Is Part Of: Journal of the Royal Statistical Society. Volume 79:Number 1(2017)
• Page Start: 51
• Page End: 67
• Publication Date: 2015-12-31
• Subjects: Goodness of fit -- High dimensional data -- Noise variance estimator -- Number of principal components -- Probabilistic principal component analysis -- Random‐matrix theory; Statistics -- Periodicals; Great Britain -- Statistics -- Periodicals; 519.2
• DOI: 10.1111/rssb.12153
• Languages: English
• ISSNs: 1369-7412

Abstract

Summary: We develop new statistical theory for probabilistic principal component analysis models in high dimensions. The focus is the estimation of the noise variance, which is an important and unresolved issue when the number of variables is large in comparison with the sample size. We first unveil the reasons for an observed downward bias of the maximum likelihood estimator of the noise variance when the data dimension is high. We then propose a bias‐corrected estimator by using random‐matrix theory and establish its asymptotic normality. The superiority of the new and bias‐corrected estimator over existing alternatives is checked by Monte Carlo experiments with various combinations of ( p, n ) (the dimension and sample size). Next, we construct a new criterion based on the bias‐corrected estimator to determine the number of the principal components, and a consistent estimator is obtained. Its good performance is confirmed by a simulation study and real data analysis. The bias‐corrected estimator is also used to derive new asymptotics for the related goodness‐of‐fit statistic under the high dimensional scheme.