Inference for Continuous Time Random Maxima with Heavy-Tailed Waiting Times
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In many complex systems of interest, inter-arrival times between events such as earthquakes, trades and neuron voltages have a heavy-tailed distribution. The set of event times is fractal-like, being dense in some time windows and empty in others, a phenomenon dubbed "bursty" in the physics literature. Renewal processes with heavy-tailed waiting times reproduce this bursty behaviour. This article develops an inference method for "Continuous Time Random Maxima" (also called "Max-renewal processes"), which assume i.i.d. magnitudes at the renewal events and model the largest cumulative magnitude. For high thresholds and infinite-mean waiting times, we show that the times between threshold crossings are Mittag-Leffler distributed, i.e. form a fractional Poisson Process. Exceedances of thresholds are known to be Generalized Pareto distributed, according to the Peaks Over Threshold approach. We model threshold crossing times and threshold exceedances jointly and provide graphical means of estimating model parameters. We show that these methods yield meaningful insights on real-world datasets.
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