Likelihood Ratio Tests for Principal Components

A particular class of tests for the principal components of a scatter matrix Σ is proposed. In the simplest case one wants to test whether a given vector is an eigenvector of Σ corresponding to its largest eigenvalue. The test statistics are likelihood ratio statistics for the classical Wishart model, and critical values are obtained parametrically as well as nonparametrically without making any assumptions on the eigenvalues of Σ. Still, the tests have asymptotic properties similar to those of classical procedures and are asymptotically admissible and optimal in some sense.