This paper provides parametric and rank-based optimal tests for eigenvectors and eigenvalues of covariance or scatter matrices in elliptical families. The parametric tests extend the Gaussian likelihood ratio tests of Anderson (1963) and their pseudo-Gaussian robustifications by Davis (1977) and Tyler (1981, 1983). The rank-based tests address a much broader class of problems, where covariance matrices need not exist and principal components are associated with more general scatter matrices. The proposed tests are shown to outperform daily practice both from the point of view of validity as from the point of view of efficiency. This is achieved by utilizing the Le Cam theory of locally asymptotically normal experiments, in the nonstandard context, however, of a curved parametrization. The results we derive for curved experiments are of independent interest, and likely to apply in other contexts.
"Optimal rank-based testing for principal components." Ann. Statist. 38 (6) 3245 - 3299, December 2010. https://doi.org/10.1214/10-AOS810