Inference for Heavy-Tailed Max-Renewal Processes
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Max-renewal processes, or Continuous Time Random Maxima, assume that events arrive according to a renewal process, and track the running maximum of the magnitudes of events up to time t. In many complex systems of interest, notably earthquakes, trades and neuron voltages, inter-arrival times exhibit heavy-tailed distributions. The dynamics of events then exhibits memory, which affects the rate at which events occur: rates are highly variable in some time intervals, while other intervals have long quiescent periods, a behaviour which has been dubbed "bursty" in the physics literature. This article provides a statistical model for the exceedances X(ℓ) and interexceedance times T(ℓ) of events whose magnitude exceeds a given threshold ℓ. We derive limit theorems for the distribution of X(ℓ) and T(ℓ) as ℓ approaches extreme values. The standard Peaks Over Threshold inference approach in extreme value theory is based on the fact that X(ℓ) is approximately generalized Pareto distributed, and models the threshold crossing times as a standard Poisson process. We show that for waiting times with infinite mean, the threshold crossing times approach a fractional Poisson process in the limit of high thresholds. The inter-arrival times of threshold crossings scale with p^1/β, where p is the threshold crossing probability and β∈(0,1) is the tail parameter for the waiting times. We provide graphical means of estimating model parameters, and show that these methods provide useful results on simulated and real-world datasets.
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