Highly robust estimation of dispersion matrices

In this paper, we propose a new componentwise estimator of a dispersion mat rix. based on a highly robust estimator of scale, The key idea is the elimi nation of a location estimator in the dispersion estimation procedure. The robustness properties are studied by means of the influence function and th e breakdown point. Further characteristics such as asymptotic variance and efficiency are also analyzed. It is shown in the componentwise approach, fo r 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 d ependence, whereas it decreases with increasing dependence for correlation estimators. We also prove that the asymptotic variance of dispersion estima tors for multivariate Gaussian distributions is proportional to the asympto tic variance of the underlying scale estimator. The proportionality value d epends only, on the underlying dependence. Therefore, the highly robust dis persion estimator is among the best robust choice at the present time in th e componentwise approach. because it is location-free and combines small va riability and robustness properties such as high breakdown point and bounde d 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 estima tor is performed. The highly robust properties of the new estimator are con firmed. A second comparison with global estimators such as the method of mo ment estimator, the minimum volume ellipsoid, and the minimum covariance de terminant estimator is also performed, with two types of outliers. In this case, the highly robust dispersion matrix estimator turns out to be an inte resting compromise between the high efficiency of the method of moment esti mator in noncontaminated situations and the highly robust properties of the minimum volume ellipsoid and minimum covariance determinant estimators in contaminated situations. (C) 2001 Academic Press.