Perturbation Inequalities and Confidence Sets for Functions of a Scatter Matrix

LetΣbe an unknown covariance matrix. Perturbation (in)equalities are derived for various scale-invariant functionals ofΣsuch as correlations (including partial, multiple and canonical correlations) or angles between eigenspaces. These results show that a particular confidence set forΣis canonical if one is interested in simultaneous confidence bounds for these functionals. The confidence set is based on the ratio of the extreme eigenvalues ofΣ−1S, whereSis an estimator forΣ. Asymptotic considerations for the classical Wishart model show that the resulting confidence bounds are substantially smaller than those obtained by inverting likelihood ratio tests.