Optimal robust estimates using the Kullback-Leibler divergence

VÍCTOR J. YOHAI

Parma Italia

Conferencia; International Conference on Robust Statistics; 2009

Universidad de Parma,

Stahel (1981) derived  optimal M-estimates that maximize a measure of efficiency  given bound the fross error sensitivity (GES) for   two invariant definitions of  GES.   However, it is not always clear which is the most appropiate standardization for a particular application. These results can be found also in Chapter 4 of Hampel et al. (1986). To overcome this problem, in this paper we propose optimal M-estimates which use measures of robustness and efficiency based on the Kullback-Leibler (KL) divergence and on the Hellinger (H) distance . The advantage of this approach is that these measures do not depend on the particular parametrization of the family of distributions, and as a consequence, the resulting optimal estimates are equivariant. We show that the optimal estimates obtained using this approach coincide with the optimal, M-estimates obtained when the GES is standardized using the information matrix.