Information Geometry and Asymptotics for Kronecker Covariances

TL;DR

This paper investigates the information geometry and asymptotic properties of estimators for Kronecker-structured covariances, proposing an efficient rescaled estimator and demonstrating high-dimensional consistency, even with minimal sample sizes.

Contribution

It introduces an asymptotically efficient rescaled partial trace estimator and an orthogonal parameterization for Kronecker covariances, advancing high-dimensional estimation techniques.

Findings

The partial trace estimator is asymptotically inefficient.

A rescaled partial trace estimator achieves asymptotic efficiency.

Consistent estimation is possible with minimal samples in high dimensions.

Abstract

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.