# Asymptotic Distribution of the Score Test for Detecting Marks in Hawkes Processes

## TL;DR

This paper derives the asymptotic distribution of the score test for detecting the influence of marks in Hawkes processes, establishing chi-squared and non-central chi-squared distributions under null and local alternatives.

## Contribution

It provides the first rigorous derivation of the asymptotic distribution of the score test for marks in Hawkes processes, including local power analysis.

## Key findings

- Score test statistic follows a chi-squared distribution under null hypothesis.
- Under local alternatives, the distribution is a non-central chi-squared.
- Results are applicable to exponential decay Hawkes processes.

## Abstract

The asymptotic distribution of the score test of the null hypothesis that marks do not impact the intensity of a Hawkes marked self-exciting point process is shown to be chi-squared. For local asymptotic power, the distribution against local alternatives is also established as non-central chi-squared. These asymptotic results are derived using existing asymptotic results for likelihood estimates of the unmarked Hawkes process model together with mild additional conditions on the moments and ergodicity of the marks process and an additional uniform boundedness assumption, shown to be true for the exponential decay Hawkes process.