The ordinary MLE is asymptotically normal for locally stationary Hawkes processes, and a Bernstein-polynomial likelihood ratio test detects any non-constant continuous time-dependent reproduction rate when the degree is chosen large enough.
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A non-local estimator for locally stationary Hawkes processes
The ordinary MLE is asymptotically normal for locally stationary Hawkes processes, and a Bernstein-polynomial likelihood ratio test detects any non-constant continuous time-dependent reproduction rate when the degree is chosen large enough.