We consider the consistency and weak convergence of $S$-estimators in the linear regression model. Sufficient conditions for consistency with varying dimension are given which are sufficiently weak to cover, for example, polynomial trends and i.i.d. carriers. A weak convergence theorem for the Hampel-Rousseeuw least median of squares estimator is obtained, and it is shown under rather general conditions that the correct norming factor is $n^{1/3}$. Finally, the asymptotic normality of $S$-estimators with a smooth $\rho$-function is obtained again under weak conditions on the carriers.
Publié le : 1990-12-14
Classification:
$S$-estimators,
linear regression,
least median of squares,
consistency,
weak convergence,
asymptotic normality,
62J05,
62F12,
62F35
@article{1176347871,
author = {Davies, Laurie},
title = {The Asymptotics of $S$-Estimators in the Linear Regression Model},
journal = {Ann. Statist.},
volume = {18},
number = {1},
year = {1990},
pages = { 1651-1675},
language = {en},
url = {http://dml.mathdoc.fr/item/1176347871}
}
Davies, Laurie. The Asymptotics of $S$-Estimators in the Linear Regression Model. Ann. Statist., Tome 18 (1990) no. 1, pp. 1651-1675. http://gdmltest.u-ga.fr/item/1176347871/