Hawkes process and Edgeworth expansion with application to maximum likelihood estimator

TL;DR

This paper develops a rigorous mathematical framework for the higher-order asymptotic behavior of the Hawkes process, providing a second-order distribution for the maximum likelihood estimator and supporting it with simulation results.

Contribution

It introduces a higher-order asymptotic theory for the Hawkes process and derives the second-order distribution of the MLE, addressing limitations of normal approximation in short observation scenarios.

Findings

Second-order asymptotic distribution for the MLE of Hawkes process.

Mathematical foundation for higher-order behavior of Hawkes process.

Simulation results validating theoretical findings.

Abstract

The Hawks process is a point process with a self-exciting property. It has been used to model earthquakes, social media events, infections, etc., and is getting a lot of attention. However, as a real problem, there are often situations where we can not obtain data with sufficient observation time. In such cases, it is not appropriate to approximate the error distribution of an estimator by the normal distribution. To overcome this problem, we provide a rigorous mathematical foundation of the theory for the higher-order asymptotic behavior of the one-dimensional Hawkes process with an exponential kernel. As an important application, we give the second-order asymptotic distribution for the maximum likelihood estimator of the exponential Hawkes process. Furthermore, we also present the simulation results.