  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 quadratic form or partial trace
estimator proposed by Linton and Tang. 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 sub-blocks of the sample
covariance matrix optimally. To correct for this, an asymptotically efficient,
rescaled partial trace estimator is proposed. 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 when the sample size is one.
