Maximum likelihood covariance matrix estimation from two possibly mismatched data sets

We consider estimating the covariance matrix from two data sets, one whose covariance matrix R-1 is the sought one and another set of samples whose covariance matrix R-2 slightly differs from the sought one, due e.g. to different measurement configurations. We assume however that the two matrices are rather close, which we formulate by assuming that R-1(1/2) (R2-1R11/2)vertical bar R-1 follows a Wishart distribution around the identity matrix. It turns out that this assumption results in two data sets with different marginal distributions, hence the problem becomes that of covariance matrix estimation from two data sets which are distribution-mismatched. The maximum likelihood estimator (MLE) is derived and is shown to depend on the values of the number of samples in each set. We show that it involves whitening of one data set by the other one, shrinkage of eigenvalues and colorization, at least when one data set contains more samples than the size p of the observation space. When both data sets have less than p samples but the total number is larger than p, the MLE again entails eigenvalues shrinkage but this time after a projection operation. Simulation results compare the new estimator to state of the art techniques. (C) 2019 Elsevier B.V. All rights reserved.