This document explores and develops methods for forecasting extreme outcomes, such as the maximum of a sample of n independent and identically distributed random variables. I was inspired to write this by Jaime Sevilla’s recent post with research ideas in forecasting and, in particular, his suggestion to write an accessible introduction to the Fisher–Tippett–Gnedenko Theorem.
I’m very grateful to Jaime Sevilla for proposing this idea and for providing great feedback on a draft of this document.
The Fisher–Tippett–Gnedenko Theorem is similar to a central limit theorem, but for the maximum of random variables. Whereas central limit theorems tell us about what happens on average, the Fisher–Tippett–Gnedenko Theorem tells us what happens in extreme cases. This makes it especially useful in risk management, when we need to pay particular attention to worst case outcomes. It could be a useful tool for forecasting tail events.
This document introduces the theorem, describes the limiting probability distribution and provides a couple of examples to illustrate the use (and misuse!) of the Fisher–Tippett–Gnedenko Theorem for forecasting. In the process, I introduce a tool that computes the distribution of the maximum n iid random variables that follow a normal distribution centrally but with an (optional) right Pareto tail.
I expect the time-poor reader to get most of the value from this document by reading the informal statement of the Fisher–Tippett–Gnedenko Theorem, the overview of the generalised extreme value distribution, and the shortest and tallest people in the world example, and then maybe making a copy and playing around with the tool for forecasting the maximum of n random variables that follow normal distributions with Pareto tails (consulting this as needed).
This is statistically neat, but I'd recommend Taleb's Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications - the most extreme cases are in practice always those for which you assumed the wrong distribution! e.g. there are many cases where a system spends most of it's time in a regime characterized by a normal distribution, and then rarely a different mechanism in the underlying dynamics shifts the whole thing wildly out of that - famously including the blowup of Long Term Capital Management after just four years.