Exponential inequalities for martingales, with application to maximum likelihood estimation for counting processes (Q1914270)

The author proves an exponential probability inequality for a special class of 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. As an application, the author considers rates of convergence for (nonparametric) maximum likelihood estimators for counting processes. A typical result from the paper is the following theorem: Let \(N = \{N_t\}_{t \geq 0}\) be a counting process with compensator \(A = \{A_t\}_{t \geq 0}\). It is assumed that \(A\) is continuous. Let \(\mathcal L\) be a class of predictable functions with \(g_t \geq -L\) for all \(t \geq 0\), \(g \in {\mathcal L}\) and for some \(0 < L < \infty\), and let \(H(\delta, b, B)\) be the entropy of \(\mathcal L\), where \(B \subset \{A_T \leq \sigma^2_t\}\). There exist constants \(C_1\), \(C_2\), \(C_3\), \(C_4\), depending on \(L\), such that for \(0 \leq \varepsilon \leq 1\) and \[ {\varepsilon b^2 \over C_1} \geq \int_{\varepsilon b^2/(C_2 \sigma_T) \wedge b/8} \sqrt {H(x,b,B)} dx \vee b, \] we have \[ \begin{multlined} P\left(\left\{ \left|\int^T_0 gd(N - A)\right |\geq \varepsilon b^2 \text{ and } \int^T_0 (e^g - 1)^2 dA \leq 2b^2 \text{ for some } g \in {\mathcal L} \right\} \cap B\right) \leq \\ \leq C_3 \text{exp} \left\{-{\varepsilon^2 b^2 \over C_4}\right\}.\end{multlined} \] {}.