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.

翻译：本文探讨了Kronecker结构协方差矩阵的估计量在n增大和p增大场景下的信息几何与渐近行为，重点分析了Linton与Tang提出的二次型或部分迹估计量。研究表明，部分迹估计量在渐近意义下非有效，其原因在于该估计量未能对样本协方差矩阵的子块进行最优缩放。为修正此问题，提出了一个渐近有效的缩放部分迹估计量。受此缩放思想的启发，我们为Kronecker协方差矩阵集合引入了一种正交参数化方法。基于部分迹估计量的高维一致性结果揭示了维数优势：当数组阶数至少为三时，随着数组各维度联合增长，即使样本量仅为1，也能对Kronecker协方差矩阵进行一致估计。