On a robust and effcient maximum depth estimator

The best breakdown point robustness is one of the most outstanding features of the univariate median.For this robustness property,the median,however,has to pay the price of a low effciency at normal and other light-tailed models.Affine equivariant multivariate analogues of the univariate median with high breakdown points were constructed in the past two decades.For the high breakdown robustness,most of them also have to sacrifice their effciency at normal and other models,nevertheless.The affine equivariant maximum depth estimator proposed and studied in this paper turns out to be an exception.Like the univariate median,it also possesses a highest breakdown point among all its multivariate competitors.Unlike the univariate median,it is also highly efficient relative to the sample mean at normal and various other distributions,overcoming the vital low-effciency shortcoming of the univariate and other multivariate generalized medians.The paper also studies the asymptotics of the estimator and establishes its limit distribution without symmetry and other strong assumptions that are typically imposed on the underlying distribution.