A parameter expressed as a functional $T(F)$ of a distribution function (df) $F$ may be estimated by the "statistical function" $T(F_n)$ based on the sample df $F_n$. For analysis of the estimation error $T(F_n) - T(F)$, we adapt the differential approach of von Mises (1947) to exploit stochastic properties of the Kolmogorov-Smirnov distance $\sup_x|F_n(x) - F(x)|$. This leads directly to the central limit theorem (CLT) and law of the iterated logarithm (LIL) for $T(F_n) - T(F)$. The adaptation also incorporates innovations designed to broaden the scope of statistical application of the concept of differential. Application to a wide class of robust-type $M$-estimates is carried out.
Publié le : 1980-05-14
Classification:
Differentials,
functionals,
statistical functions,
asymptotic normality,
law of the iterated logarithm,
$M$-estimates,
62E20,
62G35
@article{1176345012,
author = {Boos, Dennis D. and Serfling, R. J.},
title = {A Note on Differentials and the CLT and LIL for Statistical Functions, with Application to $M$-Estimates},
journal = {Ann. Statist.},
volume = {8},
number = {1},
year = {1980},
pages = { 618-624},
language = {en},
url = {http://dml.mathdoc.fr/item/1176345012}
}
Boos, Dennis D.; Serfling, R. J. A Note on Differentials and the CLT and LIL for Statistical Functions, with Application to $M$-Estimates. Ann. Statist., Tome 8 (1980) no. 1, pp. 618-624. http://gdmltest.u-ga.fr/item/1176345012/