We consider asymptotically normal statistics which are symmetric functions of N i.i.d. random variables. For these statistics we prove the validity of an Edgeworth expansion with remainder $O(N^{-1})$ under
Cramér's condition on the linear part of the statistic and moment assumptions for all parts of the statistic. By means of a counterexample we show that it is generally not possible to obtain an Edgeworth expansion with remainder $o(N^{-1})$ without imposing additional assumptions on the structure of the nonlinear part of the statistic.
Publié le : 1997-04-14
Classification:
Asymptotic expansion,
Edgeworth expansions,
symmetric statistics,
Hoeffdings's decomposition,
$U$-statistics,
functions of sample means,
functionals of empirical distribution functions,
linear combinations of order statistics,
Student's statistic,
62E20,
60F05
@article{1031833676,
author = {Bentkus, V. and G\"otze, F. and van Zwet, W. R.},
title = {An Edgeworth expansion for symmetric statistics},
journal = {Ann. Statist.},
volume = {25},
number = {6},
year = {1997},
pages = { 851-896},
language = {en},
url = {http://dml.mathdoc.fr/item/1031833676}
}
Bentkus, V.; Götze, F.; van Zwet, W. R. An Edgeworth expansion for symmetric statistics. Ann. Statist., Tome 25 (1997) no. 6, pp. 851-896. http://gdmltest.u-ga.fr/item/1031833676/