We obtain an exponential probability inequality for martingales and a uniform probability inequality for the process $\int g dN$, where $N$ is a counting process and where $g$ varies within a class of predictable functions $\mathscr{G}$. For the latter, we use techniques from empirical process theory. The uniform inequality is shown to hold under certain entropy conditions on $\mathscr{G}$. As an application, we consider rates of convergence for (nonparametric) maximum likelihood estimators for counting processes. A similar result for discrete time observations is also presented.
Publié le : 1995-10-14
Classification:
Counting process,
entropy,
exponential inequality,
Hellinger process,
martingale,
maximum likelihood,
rate of convergence,
60E15,
62G05
@article{1176324323,
author = {van de Geer, Sara},
title = {Exponential Inequalities for Martingales, with Application to Maximum Likelihood Estimation for Counting Processes},
journal = {Ann. Statist.},
volume = {23},
number = {6},
year = {1995},
pages = { 1779-1801},
language = {en},
url = {http://dml.mathdoc.fr/item/1176324323}
}
van de Geer, Sara. Exponential Inequalities for Martingales, with Application to Maximum Likelihood Estimation for Counting Processes. Ann. Statist., Tome 23 (1995) no. 6, pp. 1779-1801. http://gdmltest.u-ga.fr/item/1176324323/