TY - JOUR
T1 - Asymptotic distribution of the score test for detecting marks in hawkes processes
AU - Clinet, Simon
AU - Dunsmuir, William T. M.
AU - Peters, Gareth W.
AU - Richards, Kylie-Anne
N1 - Funding Information:
K-A Richards gratefully acknowledges PhD scholarship support by Boronia Capital Pty. Ltd., Sydney, Australia. The research of S. Clinet is supported by a special grant from Keio University. W. T.M. Dunsmuir was supported by travel funds from the Faculty of Sciences, University of New South Wales. The authors thank the referees for comments and suggestions that have improved the clarity and scope of the paper.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature B.V.
PY - 2021/10
Y1 - 2021/10
N2 - 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.
AB - 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.
KW - Ergodicity
KW - Inferential statistics
KW - Local power
KW - Marked Hawkes point process
KW - Quasi likelihood
KW - Score test
UR - http://www.scopus.com/inward/record.url?scp=85105995654&partnerID=8YFLogxK
U2 - 10.1007/s11203-021-09245-5
DO - 10.1007/s11203-021-09245-5
M3 - Article
AN - SCOPUS:85105995654
SN - 1387-0874
VL - 24
SP - 635
EP - 668
JO - Statistical Inference for Stochastic Processes
JF - Statistical Inference for Stochastic Processes
IS - 3
ER -