Improved Estimation of Eigenvalues and Eigenvectors of Covariance Matrices Using Their Sample Estimates

The problem of estimating the eigenvalues and eigenvectors of the covariance matrix associated with a multivariate stochastic process is considered. The focus is on finite sample size situations, whereby the number of observations is limited and comparable in magnitude to the observation dimension. Using tools from random matrix theory, and assuming a certain eigenvalue splitting condition, new estimators of the eigenvalues and eigenvectors of the covariance matrix are derived, that are shown to be consistent in a more general asymptotic setting than the traditional one. Indeed, these estimators are proven to be consistent, not only when the sample size increases without bound for a fixed observation dimension, but also when the observation dimension increases to infinity at the same rate as the sample size. Numerical evaluations indicate that the estimators have an excellent performance in small sample size scenarios, where the observation dimension and the sample size are comparable in magnitude.