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

We obtain an exponential probability inequality for martingales and a uniform probability inequality for the process ∫ g dN, where N is a counting process and where g varies within a class of predictable functions G. For the latter, we use techniques from empirical process theory. The uniform inequality is shown to hold under certain entropy conditions on 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.