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March, 1978 On Almost Sure Expansions for $M$-Estimates
Raymond J. Carroll
Ann. Statist. 6(2): 314-318 (March, 1978). DOI: 10.1214/aos/1176344126

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 $T_n$.

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Raymond J. Carroll. "On Almost Sure Expansions for $M$-Estimates." Ann. Statist. 6 (2) 314 - 318, March, 1978. https://doi.org/10.1214/aos/1176344126

Information

Published: March, 1978
First available in Project Euclid: 12 April 2007

zbMATH: 0376.62034
MathSciNet: MR464470
Digital Object Identifier: 10.1214/aos/1176344126

Subjects:
Primary: 62G35
Secondary: 62E20

Keywords: $M$-estimators , Invariance principles , robustness , studentization

Rights: Copyright © 1978 Institute of Mathematical Statistics

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Vol.6 • No. 2 • March, 1978
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