It is well known that the limit distribution of the supremum of the empirical distribution function $F_n$ centered at its expectation $F$ and standardized by division by its standard deviation is degenerate, if the supremum is taken on too large regions $\varepsilon_n < F(u) < \delta_n$. So it is natural to look for sequences of linear transformations, so that for given sequences of sup-regions $(\varepsilon_n, \delta_n)$ the limit of the transformed sup-statistics is nondegenerate. In this paper a partial answer is given to this problem, including the case $\varepsilon_n \equiv 0, \delta_n \equiv 1$. The results are also valid for the Studentized version of the above statistic, and the corresponding two-sided statistics are treated, too.
Publié le : 1979-01-14
Classification:
Standardized empirical distribution function,
normalized sample quantile process,
extreme value distribution,
boundary crossing of empirical process,
Poisson process,
Ornstein-Uhlenbeck process,
normalized Brownian bridge process,
goodness of fit test,
tail estimation,
62E20,
60F05
@article{1176344558,
author = {Jaeschke, D.},
title = {The Asymptotic Distribution of the Supremum of the Standardized Empirical Distribution Function on Subintervals},
journal = {Ann. Statist.},
volume = {7},
number = {1},
year = {1979},
pages = { 108-115},
language = {en},
url = {http://dml.mathdoc.fr/item/1176344558}
}
Jaeschke, D. The Asymptotic Distribution of the Supremum of the Standardized Empirical Distribution Function on Subintervals. Ann. Statist., Tome 7 (1979) no. 1, pp. 108-115. http://gdmltest.u-ga.fr/item/1176344558/