Ridge-penalised spectral least-squares estimation for point processes

TL;DR

This paper introduces a Ridge-penalised spectral least-squares estimation method for second-order stationary point processes, enabling effective parameter estimation from a single realization without extensive cross-validation.

Contribution

It proposes two novel approaches: a p-thinning-based cross-validation method and a spectral least-squares contrast, tailored for single realization point process data.

Findings

Effective parameter estimation on short observation windows

Outperforms traditional methods in simulation studies

Applicable to linear Hawkes processes

Abstract

Penalised estimation methods for point processes usually rely on a large amount of independent repetitions for cross-validation purposes. However, in the case of a single realisation of the process, existing cross-validation methods may be impractical depending on the chosen model. To overcome this issue, this paper presents a Ridge-penalised spectral least-squares estimation method for second-order stationary point processes. This is achieved through two novel approaches: a p-thinning-based cross-validation method to tune the penalisation parameter, relying on the spectral representation of the process; and the introduction of a spectral least-squares contrast based around the asymptotic properties of the periodogram of the sample. The proposed method is then illustrated by a simulation study on linear Hawkes processes in the context of parametric estimation, highlighting its performances against more traditional approaches, specifically when working with short observation windows.