Cross-validation of covariance structures using the Frobenius matrix distance as a discrepancy function

A new cross-validation criterion for selecting covariance structures in multivariate analysis is introduced. The criterion relies on the use of the Frobenius matrix distance as discrepancy function between structure-based and sample covariance matrices. Its implementation only requires to specify i) the candidate covariance structures, and ii) an arbitrary structure-dependent estimator of the population covariance whose performance is assumed to be of primary interest. No further statistical specification is needed, which makes its use feasible in a wide range of practical situations. The cross- validatory choice through the minimization of this criterion has a simple statistical interpretation, as seeking the covariance structure for which the given estimator achieves minimum (generalized) mean squared error. Simulation results are presented to show the performance of the introduced method in a variety of factor covariance structures, including settings with different eigenvalues, sample sizes and numbers of factors.