Preparing Students for the Future: Extreme Events and Power Tails

Abstract

We provide tools for identification and exploration of data with very large variability having power law tails. Such data describe extreme features of processes such as fire losses, flood, drought, financial gain/loss, hurricanes, population of cities, among others. Prediction and quantification of extreme events are at the forefront of the current research needs, as these events have the strongest impact on our lives, safety, economics, and the environment. We concentrate on the intuitive, rather than rigorous mathematical treatment of models with heavy tails. Our goal is to introduce instructors to these important models and provide some tools for their identification and exploration. The methods we provide may be incorporated into courses such as probability, mathematical statistics, statistical modeling or regression methods. Our examples come from ecology and census fields. Supplementary materials for this article are available online.

Keywords:

• Extreme events
• Heavy tail distribution
• Law of large numbers
• Power law distribution

Supplementary Materials

In the Appendix, we provide some resources for the instructors interested in sharing heavy tailed models with their students. They include: (1) representations of the Pareto and Generalized Pareto Distributions (GPD) which are useful for the simulation of heavy tailed data, (2) examples of further reading on the heavy tail distributions, and (3) R code for the graphs used in this paper. In addition, we provide R code for simulating samples from the Pareto and Generalized Pareto Distributions (GPD) and additional references.

Aknowledgments

The authors extend their thanks to the Reviewers, Editors, and Ms. Danelle Clarke for their comments and suggestions that greatly improved this manuscript.

Notes

1 Data from https://www.ncdc.noaa.gov/societal-impacts/wildfires/month/0.csv

2 Data from https://www2.census.gov/programs-surveys/popest/tables/2010-2019/cities/totals/