Modeling Extreme Events Using Heavy-Tailed Distributions

Abstract

Typically, in constructing a model for a random variable, one utilizes available samples to construct an empirical distribution function, which can then be used to estimate the probability that the random variable would exceed a prespecified threshold. However, in modeling extreme events, the threshold is often in excess of the largest sampled value observed thus far. In such cases, the use of empirical distributions would lead to the absurd conclusion that the random variable would never exceed the threshold. Therefore it becomes imperative to fit the observed samples with some appropriate distribution. For reasons explained in the paper, it is desirable to use the so-called stable distributions to fit the set of samples. In most cases, stable distributions are heavy-tailed, in that they do not have finite variance (and may not even have finite mean). However, they often do a very good job of fitting the data. This is illustrated in this paper via examples from various application areas such as finance and weather.