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In this paper, we develop new statistical theory for probabilistic principalcomponent analysis models in high dimensions. The focus is the estimation ofthe noise variance, which is an important and unresolved issue when the numberof variables is large in comparison with the sample size. We first unveil thereasons of a widely observed downward bias of the maximum likelihood estimatorof the variance when the data dimension is high. We then propose abias-corrected estimator using random matrix theory and establish itsasymptotic normality. The superiority of the new (bias-corrected) estimatorover existing alternatives is first checked by Monte-Carlo experiments withvarious combinations of $(p, n)$ (dimension and sample size). In order todemonstrate further potential benefits from the results of the paper to generalprobability PCA analysis, we provide evidence of net improvements in twopopular procedures (Ulfarsson and Solo, 2008; Bai and Ng, 2002) for determiningthe number of principal components when the respective variance estimatorproposed by these authors is replaced by the bias-corrected estimator. The newestimator is also used to derive new asymptotics for the relatedgoodness-of-fit statistic under the high-dimensional scheme.

On estimation of the noise variance in high-dimensional probabilistic principal component analysis