  We consider the problem of how many components to retain in the application
of principal component analysis when the dimension is much higher than the
number of observations. To estimate the number of components, we propose to
sequentially test skewness of the squared lengths of residual scores that are
obtained by removing leading principal components. The residual lengths are
asymptotically left-skewed if all principal components with diverging variances
are removed, and right-skewed if not. The proposed estimator is shown to be
consistent, performs well in high-dimensional simulation studies, and provides
reasonable estimates in a number of real data examples.
