Robustness properties of dispersion estimators

In this paper, we derive the influence function of dispersion estimators, based on a scale approach. The relation between the gross-error sensitivity of dispersion estimators and the one of the underlying scale estimator is described. We show that for the bivariate Gaussian distributions, the asymptotic variance of covariance estimators is minimal in the independent case, and is strictly increasing with the absolute value of the underlying covariance. The behavior of the asymptotic variance of correlation estimators seems to be the opposite, i.e. maximal for independent data, and strictly decreasing with the absolute value of the underlying correlation. In particular, dispersion estimators based on M-estimators of scale are studied closely. The one based on the median absolute deviation is the most B-robust in the class of symmetric estimators. Some other examples are analyzed, based on the maximum likelihood and the Welsch estimator of scale.