Information geometry and asymptotics for Kronecker covariances

We explore the information geometry and asymptotic behaviour of estimators for Kronecker-structured covariances, in both growing-n and growing-p scenarios, with a focus towards examining the estimator proposed by Linton and Tang, which we refer to as the partial trace estimator. It is shown that the partial trace estimator is asymptotically inefficient. An explanation for this inefficiency is that the partial trace estimator does not scale subblocks of the sample covariance matrix optimally. To correct for this, an asymptotically efficient, rescaled partial trace estimator is introduced. Motivated by this rescaling, we introduce an orthogonal parameterization for the set of Kronecker covariances. High-dimensional consistency results using the partial trace estimator are obtained that demonstrate a blessing of dimensionality. In settings where an array has at least order three, it is shown that as the array dimensions jointly increase, it is possible to consistently estimate the Kronecker covariance matrix, even whenthesamplesizeisone.

Duke Scholars

Author Peter David Hoff Statistical Science