Optimal robust estimates using the Kullback–Leibler divergence

We define two measures of the performance of an estimating functional TT of a multi-dimensional parameter, based on the Kullback–Leibler (KL) divergence. The first one is the KL sensitivity which measures the degree of robustness of the estimate under infinitesimal outlier contamination and the second one is the KL efficiency, which measures the asymptotic efficiency of the estimate based on TT when the assumed model holds. Using these two measures we define optimal robust M-estimates using the Hampel approach. The optimal estimates are defined by maximizing the KL efficiency subject to a bound on the KL sensitivity. In this paper we show that these estimates coincide with the optimal estimates corresponding to another Hampel problem studied by Stahel [Stahel, W.A., 1981. Robust estimation, infinitesimal optimality and covariance matrix estimators. Ph.D. Thesis, ETH, Zurich]: to minimize the trace of a standardized asymptotic covariance matrix subject to a bound on the norm of a standardized gross error sensitivity, where both the asymptotic covariance matrix and the gross error sensitivity are standardized by means of the Fisher information matrix.