We show that the limiting variance of a sequence of estimators for a structured covariance matrix has a general form that appears as the variance of a scaled projection of a random matrix that is of radial type and a similar result is obtained for the corresponding sequence of estimators for the vector of variance components. These results are illustrated by the limiting behavior of estimators for a linear covariance structure in a variety of multivariate statistical models. We also derive a characterization for the influence function of corresponding functionals. Furthermore, we derive the limiting distribution and influence function of scale invariant mappings of such estimators and their corresponding functionals. As a consequence, the asymptotic relative efficiency of different estimators for the shape component of a structured covariance matrix can be compared by means of a single scalar and the gross error sensitivity of the corresponding influence functions can be compared by means of a single index. Similar results are obtained for estimators of the normalized vector of variance components. We apply our results to investigate how the efficiency, gross error sensitivity, and breakdown point of S-estimators for the normalized variance components are affected simultaneously by varying their cutoff value.

翻译：我们证明了结构化协方差矩阵估计量序列的极限方差具有一种通用形式，该形式表现为径向型随机矩阵的缩放投影的方差，并且对于方差分量向量的相应估计量序列也得到了类似结果。这些结果通过多种多元统计模型中线性协方差结构估计量的极限行为得到说明。我们还推导了相应泛函影响函数的特征表达式。此外，我们建立了此类估计量及其对应泛函的尺度不变映射的极限分布与影响函数。由此，结构化协方差矩阵形状分量不同估计量的渐近相对效率可通过单一标量进行比较，而相应影响函数的总体误差敏感度可通过单一指标进行对比。对于归一化方差分量向量的估计量也获得了类似结论。我们应用所得结果研究了S估计量在归一化方差分量估计中，其效率、总体误差敏感度和崩溃点如何随截断值的变化而同时受到影响。