  Principal component analysis (PCA) is a well-known tool in multivariate
statistics. One significant challenge in using PCA is the choice of the number
of components. In order to address this challenge, we propose an exact
distribution-based method for hypothesis testing and construction of confidence
intervals for signals in a noisy matrix. Assuming Gaussian noise, we use the
conditional distribution of the singular values of a Wishart matrix and derive
exact hypothesis tests and confidence intervals for the true signals. Our paper
is based on the approach of Taylor, Loftus and Tibshirani (2013) for testing
the global null: we generalize it to test for any number of principal
components, and derive an integrated version with greater power. In simulation
studies we find that our proposed methods compare well to existing approaches.
