Statistical Inference for Networks of High-Dimensional Point Processes
This addresses a critical gap in scientific applications like neuroscience, where evaluating uncertainty in network interactions is essential, though it is incremental by building on existing estimation methods.
The paper tackles the problem of uncertainty quantification for network estimates in high-dimensional Hawkes processes, developing a new statistical inference procedure that is validated through simulations and applied to neuron spike train data.
Fueled in part by recent applications in neuroscience, the multivariate Hawkes process has become a popular tool for modeling the network of interactions among high-dimensional point process data. While evaluating the uncertainty of the network estimates is critical in scientific applications, existing methodological and theoretical work has primarily addressed estimation. To bridge this gap, this paper develops a new statistical inference procedure for high-dimensional Hawkes processes. The key ingredient for this inference procedure is a new concentration inequality on the first- and second-order statistics for integrated stochastic processes, which summarize the entire history of the process. Combining recent results on martingale central limit theory with the new concentration inequality, we then characterize the convergence rate of the test statistics. We illustrate finite sample validity of our inferential tools via extensive simulations and demonstrate their utility by applying them to a neuron spike train data set.