Asymptotic distribution of the score test for detecting marks in hawkes processes

Simon Clinet, William T. M. Dunsmuir*, Gareth W. Peters, Kylie-Anne Richards

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


The score test is a computationally efficient method for determining whether marks have a significant impact on the intensity of a Hawkes process. This paper provides theoretical justification for use of this test. It is shown that the score statistic has an asymptotic chi-squared distribution under the null hypothesis that marks do not impact the intensity process. For local power, the asymptotic distribution against local alternatives is proved to be non-central chi-squared. A stationary marked Hawkes process is constructed using a thinning method when the marks are observations on a continuous time stationary process and the joint likelihood of event times and marks is developed for this case, substantially extending existing results which only cover independent and identically distributed marks. These asymptotic chi-squared distributions required for the size and local power of the score test extend existing asymptotic results for likelihood estimates of the unmarked Hawkes process model under 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.

Original languageEnglish
Pages (from-to)635-668
Number of pages34
JournalStatistical Inference for Stochastic Processes
Issue number3
Early online date19 May 2021
Publication statusPublished - Oct 2021


  • Ergodicity
  • Inferential statistics
  • Local power
  • Marked Hawkes point process
  • Quasi likelihood
  • Score test

ASJC Scopus subject areas

  • Statistics and Probability


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