On Almost Sure Expansions for $M$-Estimates

Carroll, Raymond J.

Ann. Statist., Tome 6 (1978) no. 1, p. 314-318 / Harvested from Project Euclid

Résumé

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 $T_n$.

Détail

Publié le : 1978-03-14
Classification:  $M$-estimators,  robustness,  Studentization,  invariance principles,  62G35,  62E20