On the Asymptotic Relation Between $L$-Estimators and $M$-Estimators and their Asymptotic Efficiency Relative to the Cramer-Rao Lower Bound

Abstract

This paper gives conditions under which $L$- and $M$-estimators of a location parameter have asymptotically the same distribution, conditions under which they are asymptotically equivalent and conditions under which $L$-estimators are asymptotically efficient relative to the Cramer-Rao lower bound. Our results differ from analogous results of Jung (1955), Bickel (1965), Chernoff, Gastwirth, Johns (1967), Jaeckel (1971) and Rivest (1978, 1982) in that we do not require that the derivative of the density of the observations is absolutely continuous nor that the function defining the $M$-estimator is absolutely continuous.