Penalized Estimation of Covariance Matrices with Flexible Amounts of Shrinkage

Penalized maximum likelihood estimation has been advocated for its capability to yield substantially improved estimates of covariance matrices, but so far only cases with equal numbers of records have been considered. We show that a generalization of the inverse Wishart distribution can be utilised to derive penalties which allow for differential penalization for different blocks of the matrices to be estimated. However, this requires multiple tuning factors to be determined and thus can increase computational requirements markedly. Simulation results are presented which indicate that the additional gains obtainable for estimates of genetic covariance components - over and above those from a simple, non-differential scheme - are moderate, even if numbers of records for different traits differ by orders of magnitude.