## The Annals of Statistics

### The Asymptotics of $S$-Estimators in the Linear Regression Model

Laurie Davies

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 18, Number 4 (1990), 1651-1675.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176347871

Digital Object Identifier
doi:10.1214/aos/1176347871

Mathematical Reviews number (MathSciNet)
MR1074428

Zentralblatt MATH identifier
0719.62042

JSTOR
Davies, Laurie. The Asymptotics of $S$-Estimators in the Linear Regression Model. Ann. Statist. 18 (1990), no. 4, 1651--1675. doi:10.1214/aos/1176347871. https://projecteuclid.org/euclid.aos/1176347871