The Annals of Statistics

Exponential Inequalities for Martingales, with Application to Maximum Likelihood Estimation for Counting Processes

Sara van de Geer

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Abstract

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.

Article information

Source
Ann. Statist., Volume 23, Number 5 (1995), 1779-1801.

Dates
First available in Project Euclid: 11 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176324323

Digital Object Identifier
doi:10.1214/aos/1176324323

Mathematical Reviews number (MathSciNet)
MR1370307

Zentralblatt MATH identifier
0852.60019

JSTOR
links.jstor.org

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62G05: Estimation

Keywords
Counting process entropy exponential inequality Hellinger process martingale maximum likelihood rate of convergence

Citation

van de Geer, Sara. Exponential Inequalities for Martingales, with Application to Maximum Likelihood Estimation for Counting Processes. Ann. Statist. 23 (1995), no. 5, 1779--1801. doi:10.1214/aos/1176324323. https://projecteuclid.org/euclid.aos/1176324323


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