Highly Robust Estimation of Dispersion Matrices

In this paper, we propose a new componentwise estimator of a dispersion matrix, based on a highly robust estimator of scale. The key idea is the elimination of a location estimator in the dispersion estimation procedure. The robustness properties are studied by means of the influence function and the breakdown point. Further characteristics such as asymptotic variance and efficiency are also analyzed. It is shown in the componentwise approach, for multivariate Gaussian distributions, that covariance matrix estimation is more difficult than correlation matrix estimation. The reason is that the asymptotic variance of the covariance estimator increases with increasing dependence, whereas it decreases with increasing dependence for correlation estimators. We also prove that the asymptotic variance of dispersion estimators for multivariate Gaussian distributions is proportional to the asymptotic variance of the underlying scale estimator. The proportionality value depends only on the underlying dependence. Therefore, the highly robust dispersion estimator is among the best robust choice at the present time in the componentwise approach, because it is location-free and combines small variability and robustness properties such as high breakdown point and bounded influence function. A simulation study is carried out in order to assess the behavior of the new estimator. First, a comparison with another robust componentwise estimator based on the median absolute deviation scale estimator is performed. The highly robust properties of the new estimator are confirmed. A second comparison with global estimators such as the method of moment estimator, the minimum volume ellipsoid, and the minimum covariance determinant estimator is also performed, with two types of outliers. In this case, the highly robust dispersion matrix estimator turns out to be an interesting compromise between the high efficiency of the method of moment estimator in noncontaminated situations and the highly robust properties of the minimum volume ellipsoid and minimum covariance determinant estimators in contaminated situations.