Parameter Estimation for the Fractional Hawkes Process

We estimate the parameters of the fractional Hawkes process using maximum likelihood and approximate Bayesian computation (ABC). To this purpose, we first derive the analytical form of the likelihood function of the process. It turns out that maximum likelihood is able to estimate the parameters at a good level of accuracy. Then, with Monte Carlo simulations, we compute the posteriors and we study the efficiency and performance of ABC procedures for estimating parameters. In the absence of theorems on sufficient statistics, we use five different statistics and a uniform prior and we compare the approximate posterior distribution with the exact likelihood for each parameter. The performance is determined quantitatively using Kullback-Leibler divergence. Our results show that the ABC method does a reasonable job at capturing the posterior distributions for parameters appearing linearly in the formula for the complete intensity, namely the base rate and the reproduction number.Supplementary materials accompanying this paper appear on-line