On estimation of a partitioned covariance matrix with linearly structured blocks

The aim of this paper is to introduce an estimation method for a linearly structured partitioned covariance matrix. In contrast to well known linear structures of partitioned matrices, for example block compound symmetry, we allow the diagonal blocks of the covariance matrix to be of different dimensions. We adapt the shrinkage method to improve the properties of the projection of the sample covariance matrix onto the linear structure space. As spaces of target matrices, we choose various quadratic subspaces of structure space. This is a novel approach in the context of the structure space under consideration, and as a result a positive definite and well-conditioned estimator having the desired structure is determined. It is also shown that the statistical and algebraic properties of the estimator depend on the choice of target space.