## The Annals of Statistics

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

Sara van de Geer

#### 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

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