Estimating the Hawkes Process From a Discretely Observed Sample Path
针对离散观测下霍克斯过程似然函数难以计算的问题，提出基于状态空间表示和序贯蒙特卡洛的无偏似然估计器，结合Metropolis-Hastings算法进行参数推断，模拟和实际数据（东京麻疹周病例数）验证其均方误差更小且置信区间易得。
Estimating the Hawkes process from a discretely observed sample path is challenging due to the intractability of the likelihood in such cases. To overcome this, we employ a state-space representation of the incomplete data problem and use the sequential Monte Carlo (SMC, aka particle filters) to approximate the likelihood function. The resulting estimator of the likelihood function is unbiased and, therefore, can be used along with the Metropolis-Hastings algorithm to construct Markov Chains to approximate the likelihood distribution and, more generally, the posterior distribution of model parameters. The performance of our methodology is assessed using simulation experiments and compared with other recently published methods. The proposed estimator exhibits a smaller mean square error compared to two benchmark estimators. Furthermore, an advantage of our method compared to existing methods is that confidence intervals for the parameters are readily computable. Finally, we apply the proposed estimator to the analysis of weekly count data on measles cases in Tokyo, Japan, and compare the results to one of the benchmark methods. The online supplementary materials contain a Julia package that implements our methodology, along with the technical proofs for two propositions.