Ridge-penalised spectral least-squares estimation for point processes
Miguel Martinez Herrera, Felix Cheysson
TL;DR
This paper tackles penalised estimation for point processes when only a single realisation is available. It develops a ridge-penalised spectral least-squares estimator guided by a p-thinning cross-validation scheme and a periodogram-based contrast to recover the spectral density from data. The authors demonstrate, via simulations of linear Hawkes processes, that the method yields superior performance over traditional spectral approaches in short windows and remains broadly applicable to second-order stationary point processes. The approach provides a practical tool for estimating Hawkes-type models when conditional intensities are difficult to handle or cross-validation is otherwise unavailable.
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.