Spectral analysis for the inference of noisy Hawkes processes

Classic estimation methods for Hawkes processes rely on the assumption that observed event times are indeed a realization of a Hawkes process, without considering any perturbation of the model. In practice, observations are often altered by some noise, and so we consider, in this work, the observations to be the indistinguishable union of event times coming from a Hawkes process and from an independent Poisson process. Since standard inference methods are either unworkable or numerically prohibitive, we propose an estimation procedure based on the spectral analysis of second‐order properties of the process. Novel results include sufficient conditions for identifiability of the model: Although we mainly focus on the exponential scenario, other types of kernels are investigated. We propose a new estimator based on maximizing the spectral log‐likelihood that, besides being free from knowing the source of each observed time, is shown to perform accurately in estimating both processes.