Score test for marks in Hawkes processes

Abstract

A score statistic for detecting the impact of marks in a linear Hawkes self-exciting point process is proposed, with its asymptotic properties, finite sample performance, power properties using simulation and application to real data presented. A major advantage of the proposed inference procedure is that the Hawkes process can be fitted under the null hypothesis that marks do not impact the intensity process. Hence, for a given record of a point process, the intensity process is estimated once only and then assessed against any number of potential marks without refitting the joint likelihood each time. Marks can be multivariate and serially dependent. The score function for any given set of marks is easily constructed as the covariance of functions of future intensities fits the unmarked process with functions of the marks under assessment. The asymptotic distribution of the score statistic is a Chi-squared distribution, with degrees of freedom equal to the number of parameters required to specify the boost function. Model-based or nonparametric estimation of required features of the mark’s marginal moments and serial dependence can be used. Using sample moments of the marks in the test statistic construction does not impact the size and power properties.