The Annals of Statistics

On $M$-Processes and $M$-Estimation

A. H. Welsh

Full-text: Open access

Abstract

We relate the asymptotic behavior of $M$-estimators of the regression parameter in a linear model in which the dimension of the regression parameter may increase with the sample size to the stochastic equicontinuity of an associated $M$-process. The approach synthesises a number of results for the dimensionally fixed regression model and then extends these results in a direct unified way. The resulting theorems require only mild conditions on the $\psi$-function and the underlying distribution function. In particular, the results do not require $\psi$ to be smooth and hence can be applied to such estimators as the least absolute deviations estimator. We also treat one-step $M$-estimation.

Article information

Source
Ann. Statist., Volume 17, Number 1 (1989), 337-361.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176347021

Digital Object Identifier
doi:10.1214/aos/1176347021

Mathematical Reviews number (MathSciNet)
MR981455

Zentralblatt MATH identifier
0701.62074

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62J05: Linear regression 60F05: Central limit and other weak theorems

Keywords
Asymptotic linearity large $p$ asymptotics $M$-estimators one-step $M$-estimators regression quantiles robust estimation stochastic equicontinuity

Citation

Welsh, A. H. On $M$-Processes and $M$-Estimation. Ann. Statist. 17 (1989), no. 1, 337--361. doi:10.1214/aos/1176347021. https://projecteuclid.org/euclid.aos/1176347021


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Corrections

  • See Correction: A. H. Welsh. Correction: On $M$-Processes and $M$-Estimation. Ann. Statist., Volume 18, Number 3 (1990), 1500--1500.