Introduction to Extreme Value Theory and Its Uses

On Monday I attended three talks on the topic of Extreme Value Theory. The speakers were Prof Jon Tawn, Dr Emma Eastoe and Dr Jenny Wadsworth, all of the department of Mathematics and Statistics here at Lancaster University. Jon gave a introduction and overview of the theory, highlighting the important results and the uses/applications. Emma then went on to discuss some extreme value models for environmental processes before Jenny introduced the theory of multivariate extremes.

Motivations for the theory

The world we live in is very uncertain, which can be a real pain when you are trying to plan things for the future. Anyone working on contingency plans for post-Brexit will know what I mean. It therefore pays to have ways of measuring this uncertainty. This was the original motivation for probability theory; the building blocks for statistics. Some people would really like to know the uncertainty of very extreme events occurring, insurance companies for example. Using past data, it would be very beneficial for them to be able to estimate the 'risk' of certain events occurring, such as floods, so they can undertake the necessary measures to ensure they have enough money set aside to cover future claims. There are many more examples of where these kinds of insights are useful, such as planning for flood defenses, examining air pollution levels for public health or even space weather for satellite operators. 

An overview 

The general idea is that we only want to consider the extreme values in the data we have, i.e. we don't really need all of it. One way to do this is to split the data up into intervals, and then consider the maximal value the data takes over them. For example, in the diagram below (shown by Prof Tawn during his talk) we have some data on river flows at different times. We split this data up into years and then only focus on the annual maxima.

The question is then how do we use this data to learn something about the nature of extreme river flow values? This is where extreme value theory really comes into its own. There is an amazing result know as the Extremal Types Theorem which says

we get the same distribution for the maximum of a large number of variables whatever their original distribution.

This means that we can now fit this all-inclusive distribution to our data and then make justified statistical inferences from it. The best part is we don't have to assume the data come from any distribution in particular, which often has to be done when doing other statistical analysis.

Illustrating the Extremal Types Theorem

This may or may not sound a little hard to believe to you, it certainly did to me. Either way, it is very informative to consider an example. To do this we are going to assume we have some data from to two different underlying distributions, exponential and normal. Mathematically, lets assume we have two IID samples \(X_1, X_2, \dots , X_n\) and \(Y_1, Y_2, \dots , Y_n\) where \(X_i \sim \textrm{Exp}(1)\) and \(Y_i \sim \textrm{N}(0,1)\). Now, we are interested in the maximal value

$$M_n (\mathbf{X})= \textrm{Max}(X_1, X_2, \dots , X_n).$$

By the extremal types theorem, we expect that \(M_n (\mathbf{X})\) and \(M_n (\mathbf{Y})\) to converge to similar distributions as \(n\) gets large. You can see below that the distributions are initially very different, with the exponential zero for all negative values while the normal distribution is positive for all reals. However, in the animation below I have plotted both \(M_n (\mathbf{X})\) and \(M_n (\mathbf{Y})\) as \(n\) goes from \(n=1\) up to \(n=500\) (with some normalizing constants). One can clearly see here that we get convergence we expect!