The Annals of Statistics

On Almost Sure Expansions for $M$-Estimates

Raymond J. Carroll

Full-text: Open access

Abstract

Let $T_n$ be an $M$-estimator with defining function $\psi$ and preliminary estimate of scale $s_n$. Without loss of generality, let $s_n \rightarrow 1$ and take $E\psi(X/\xi) = 0$. Under various conditions, it is shown that any consistent version of $T_n$ is almost surely to order $O(n^{-1} \log_2 n)$ a linear combination of $n^{-1} \sum^n_1 \psi(X_i)$ and $s_n$. Only in the case $EX_1\psi'(X_1) = 0$ does the contribution of $S_n$ vanish; it is shown how this affects the estimation of the asymptotic variance of $T_n$.

Article information

Source
Ann. Statist., Volume 6, Number 2 (1978), 314-318.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176344126

Digital Object Identifier
doi:10.1214/aos/1176344126

Mathematical Reviews number (MathSciNet)
MR464470

Zentralblatt MATH identifier
0376.62034

JSTOR
links.jstor.org

Subjects
Primary: 62G35: Robustness
Secondary: 62E20: Asymptotic distribution theory

Keywords
$M$-estimators robustness Studentization invariance principles

Citation

Carroll, Raymond J. On Almost Sure Expansions for $M$-Estimates. Ann. Statist. 6 (1978), no. 2, 314--318. doi:10.1214/aos/1176344126. https://projecteuclid.org/euclid.aos/1176344126


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