Goodness-of-Fit Test for Mismatched Self-Exciting Processes

Song Wei, Shixiang Zhu, Minghe Zhang, Yao Xie

Abstract: Recently there have been many research ef- forts in developing generative models for self- exciting point processes, partly due to their broad applicability for real-world applications. However, rarely can we quantify how well the generative model captures the nature or ground-truth since it is usually unknown. The challenge typically lies in the fact that the generative models typically provide, at most, good approximations to the ground- truth (e.g., through the rich representative power of neural networks), but they cannot be precisely the ground-truth. We thus can- not use the classic goodness-of-fit (GOF) test framework to evaluate their performance. In this paper, we develop a GOF test for genera- tive models of self-exciting processes by mak- ing a new connection to this problem with the classical statistical theory of Quasi-maximum- likelihood estimator (QMLE). We present a non-parametric self-normalizing statistic for the GOF test: the Generalized Score (GS) statistics, and explicitly capture the model misspecification when establishing the asymp- totic distribution of the GS statistic. Nu- merical simulation and real-data experiments validate our theory and demonstrate the pro- posed GS test’s good performance.