Asymptotically distribution-free goodness-of-fit testing for point processes

Abstract

Consider an observed path from a multivariate temporal point process $N$ with law $\mathbb{P}$ on the time interval $[0,T]$. To test the null hypothesis that $\mathbb{P}$ belongs to a given parametric family, we construct a compensated empirical process to which we apply a suitable innovation martingale transformation. We prove that the resulting test process converges weakly to a standard Wiener process. Consequently, taking a testing functional of this process yields an asymptotically distribution-free goodness-of-fit test for parametric point processes. Furthermore, for standard test statistics based on the increments of this test process, we establish consistency under alternative hypotheses. We assess the performance of the proposed testing procedure through a Monte Carlo simulation study and illustrate its practical utility with two real-data examples.