Spectral estimation of Hawkes processes from count data

TL;DR

This paper introduces a spectral estimation method for linear stationary Hawkes processes using count data over fixed intervals, enabling consistent parameter estimation when exact event locations are unobserved.

Contribution

It adapts Whittle's spectral estimation to count data, providing a new approach for parameter inference in partially observed Hawkes processes.

Findings

Spectral estimation yields consistent and asymptotically normal estimators.

Method performs well even with large time intervals.

Case study demonstrates practical applicability.

Abstract

This paper presents a parametric estimation method for ill-observed linear stationary Hawkes processes. When the exact locations of points are not observed, but only counts over time intervals of fixed size, methods based on the likelihood are not feasible. We show that spectral estimation based on Whittle's method is adapted to this case and provides consistent and asymptotically normal estimators, provided a mild moment condition on the reproduction function. Simulated datasets and a case-study illustrate the performances of the estimation, notably of the reproduction function even when time intervals are relatively large.