Discrete Extremes

TL;DR

This paper extends extreme value analysis to discrete data by proposing two new models, the discrete generalized Pareto and generalized Zipf, which effectively estimate rare events across various real-world applications.

Contribution

It introduces two novel methods for modeling extremes in discrete data, broadening the applicability of extreme value theory.

Findings

Both models perform well in simulated data.

Effective in real-world applications like word frequency and tornado outbreaks.

Show improved estimation of rare events.

Abstract

Our contribution is to widen the scope of extreme value analysis applied to discrete-valued data. Extreme values of a random variable $X$ are commonly modeled using the generalized Pareto distribution, a method that often gives good results in practice. When $X$ is discrete, we propose two other methods using a discrete generalized Pareto and a generalized Zipf distribution respectively. Both are theoretically motivated and we show that they perform well in estimating rare events in several simulated and real data cases such as word frequency, tornado outbreaks and multiple births.