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

Selecting the number of principal components: estimation of the true rank of a noisy matrix