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Title: Alternative asymptotic inference theory for a nonstationary Hawkes process Authors:  Jeffrey Kwan - University of New South Wales (UNSW) Sydney (Australia) [presenting]
Feng Chen - UNSW Syd (Australia)
William Dunsmuir - The University of New South Wales (Australia)
Abstract: The Hawkes process is a popular point process model for events that exhibit a local clustering behaviour. It has been previously studied the asymptotic inference theory for a nonstationary Hawkes process where the baseline intensity is proportional to the sometime-varying function with the proportionality constant $n$ tending to infinity. However, it was assumed the excitation kernel to be independent of $n$, and therefore, as $n$ increases, the waiting times from a baseline event to its excited events are of order $O_P(1)$ while the waiting times between baseline events is $O_P(1/n)$, suggesting the excitation effect is not local any more, which defeats the purpose of choosing the exponential excitation kernel in the first place. To overcome this issue, we study the model in a more realistic setting where the excitation kernel also depends on the limit index $n$, so that the waiting times to excited events are of the same order ofmagnitude as those between baseline events. We establish consistency and asymptotic normality of the Maximum Likelihood Estimator, and derive the asymptotic properties of the score test. We will illustrate applications to ultra-high frequency financial data and verify the asymptotic results through simulation experiments.