Goodness-of-Fit Test for Mismatched Self-Exciting Processes
This provides a method to quantify model fit for self-exciting processes, addressing a gap in generative modeling for real-world applications like event prediction.
The paper tackles the problem of evaluating generative models for self-exiting point processes when the ground-truth is unknown, by developing a goodness-of-fit test based on Quasi-maximum-likelihood estimator theory, with validation through numerical simulations and real-data experiments showing good performance.
Recently there have been many research efforts 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 cannot use the classic goodness-of-fit (GOF) test framework to evaluate their performance. In this paper, we develop a GOF test for generative models of self-exciting processes by making 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 asymptotic distribution of the GS statistic. Numerical simulation and real-data experiments validate our theory and demonstrate the proposed GS test's good performance.