  In this paper, 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 of a widely observed downward bias of the maximum likelihood estimator
of the variance when the data dimension is high. We then propose a
bias-corrected estimator using random matrix theory and establish its
asymptotic normality. The superiority of the new (bias-corrected) estimator
over existing alternatives is first checked by Monte-Carlo experiments with
various combinations of $(p, n)$ (dimension and sample size). In order to
demonstrate further potential benefits from the results of the paper to general
probability PCA analysis, we provide evidence of net improvements in two
popular procedures (Ulfarsson and Solo, 2008; Bai and Ng, 2002) for determining
the number of principal components when the respective variance estimator
proposed by these authors is replaced by the bias-corrected estimator. The new
estimator is also used to derive new asymptotics for the related
goodness-of-fit statistic under the high-dimensional scheme.
