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

TL;DR

This paper introduces a novel goodness-of-fit test for generative self-exciting point process models, addressing the challenge of evaluating models that approximate but do not exactly match the ground-truth.

Contribution

It develops a new GOF test based on the Quasi-maximum-likelihood framework and introduces the Generalized Score statistic for assessing model fit.

Findings

The GS test effectively detects model misspecification.

Numerical simulations confirm the theoretical properties.

Real-data experiments demonstrate practical applicability.

Abstract

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.