  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.
