  Multi-group covariance estimation for matrix-variate data with small within
group sample sizes is a key part of many data analysis tasks in modern
applications. To obtain accurate group-specific covariance estimates, shrinkage
estimation methods which shrink an unstructured, group-specific covariance
either across groups towards a pooled covariance or within each group towards a
Kronecker structure have been developed. However, in many applications, it is
unclear which approach will result in more accurate covariance estimates. In
this article, we present a hierarchical prior distribution which flexibly
allows for both types of shrinkage. The prior linearly combines shrinkage
across groups towards a shared pooled covariance and shrinkage within groups
towards a group-specific Kronecker covariance. We illustrate the utility of the
proposed prior in speech recognition and an analysis of chemical exposure data.
