Likelihood-Based Inference for Event-Driven Systems using Hawkes Processes with Middle-Censored Event Times

Middle censoring occurs when observations falling within certain intervals of the observation period are unavailable, while those outside these intervals remain fully observed. Such situations arise naturally in reliability studies, communication networks, epidemiological surveillance, financial transactions, and many other event-driven systems where data collection may be interrupted or incomplete. Most existing methodologies for middle-censored data have been developed under the assumption that observations are independent. In practice, however, many event-generating mechanisms exhibit temporal dependence, in which the occurrence of one event influences the likelihood of future events, leading to clustering and self-excitation effects that independent models cannot adequately capture. Motivated by this gap, this study develops a middle-censoring framework for dependent event data based on the Hawkes process. As a selfexciting point process, the Hawkes model allows past events to directly affect future event intensities, making it well-suited for applications such as contagion modelling, information diffusion, social interactions, financial market activity, and seismic events. In the proposed framework, events occurring within randomly generated non-overlapping censoring intervals are not directly observed, resulting in incomplete event histories and creating substantial challenges for statistical inference. To address these challenges, a likelihood-based framework is developed by treating the unobserved event times as latent variables. Because the observed-data likelihood involves high-dimensional integration over the missing event configurations, direct maximisation is computationally infeasible. Therefore, several estimation approaches are explored. A stochastic expectation-maximisation (SEM) algorithm combined with data augmentation is employed for likelihood-based inference, while Bayesian methods are considered to account for parameter uncertainty. In addition, machine learning techniques are investigated as flexible data-driven alternatives for parameter estimation from partially observed event histories. The performance of these methods is evaluated and compared in terms of estimation accuracy, computational efficiency, bias, and mean squared error. Finally, extensive Monte Carlo simulation studies are conducted under varying levels of censoring and different degrees of self-excitation.