Goodness-of-Fit Testing for Point Processes in Large Populations

TL;DR

This paper introduces a new asymptotically distribution-free goodness-of-fit test for parametric point processes in large populations, applicable to various models and demonstrated through simulations and real data.

Contribution

It develops a novel unitary transformation approach that enables distribution-free testing of parametric point process models under large population asymptotics.

Findings

Good finite-sample performance demonstrated via Monte Carlo simulations.

Applicable to survival, cure, and software reliability models.

Effective with censored data and real-world examples.

Abstract

Suppose we have an observed path from a point process counting event occurrences in a large population. Based on the observed path, we would like to test the null hypothesis that the conditional intensity of the point process belongs to a particular parametric family. We propose a novel approach to conducting such goodness-of-fit tests. The idea is to construct a unitary transformation of a natural parametric testing process such that it converges weakly to a ``standard'' target process, independent of the particular parametric form assumed under the null hypothesis. This transformation therefore paves the way for asymptotically distribution-free goodness-of-fit testing of parametric point processes. We demonstrate the good finite-sample performance of our approach through Monte Carlo simulations of Aalen-type survival processes, without and with censoring, mixture cure models, and software reliability models, and we illustrate its applicability with observed human lifetimes as well as real software failures.