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.

翻译：点过程的惩罚估计方法通常依赖于大量独立重复样本进行交叉验证。然而，当仅存在过程的单次实现时，现有交叉验证方法可能因所选模型不同而难以实施。为解决这一问题，本文针对二阶平稳点过程提出了一种基于岭惩罚的谱最小二乘估计方法。该方法通过两种创新途径实现：基于p-稀释的交叉验证方法（依托过程的谱表示）来调整惩罚参数；以及基于样本周期图渐近性质构建的谱最小二乘对比函数。随后通过线性霍克斯过程参数估计场景下的模拟研究，验证了所提方法的有效性，特别在短观测窗口条件下，其性能显著优于传统方法。