## The Annals of Statistics

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

Constance van Eeden

#### 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.

#### Article information

Source
Ann. Statist., Volume 11, Number 2 (1983), 674-690.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176346172

Digital Object Identifier
doi:10.1214/aos/1176346172

Mathematical Reviews number (MathSciNet)
MR696078

Zentralblatt MATH identifier
0526.62038

JSTOR
van Eeden, Constance. On the Asymptotic Relation Between $L$-Estimators and $M$-Estimators and their Asymptotic Efficiency Relative to the Cramer-Rao Lower Bound. Ann. Statist. 11 (1983), no. 2, 674--690. doi:10.1214/aos/1176346172. https://projecteuclid.org/euclid.aos/1176346172