A one-step version $M^{(1)}_n$ and a two-step version $M^{(2)}_n$ of a general $M$-estimator $M_n$ are suggested such that $M_n - M^{(1)}_n = O_p(n^{-1})$ and $M_n - M^{(2)}_n = O_p(n^{-3/2})$ for every $n^{1/2}$-consistent initial estimator and under some regularity conditions. In the special case of maximum likelihood estimation, this among other yields that the second-order efficiency properties of $M^{(2)}_n$ coincide with those of $M_n$. An application to the Pitman estimator of location is considered.
Publié le : 1985-09-14
Classification:
$M$-estimator,
$n^{1/2}$-consistent estimator,
maximum likelihood estimator,
Pitman's estimator,
second-order asymptotic linearity,
62F12,
62G05
@article{1176349666,
author = {Janssen, P. and Jureckova, J. and Veraverbeke, N.},
title = {Rate of Convergence of One- and Two-Step $M$-Estimators with Applications to Maximum Likelihood and Pitman Estimators},
journal = {Ann. Statist.},
volume = {13},
number = {1},
year = {1985},
pages = { 1222-1229},
language = {en},
url = {http://dml.mathdoc.fr/item/1176349666}
}
Janssen, P.; Jureckova, J.; Veraverbeke, N. Rate of Convergence of One- and Two-Step $M$-Estimators with Applications to Maximum Likelihood and Pitman Estimators. Ann. Statist., Tome 13 (1985) no. 1, pp. 1222-1229. http://gdmltest.u-ga.fr/item/1176349666/