August 13, 2020. A quick introduction to extreme values.
Extreme values in politics and probability
I recently stumbled onto the beautiful lectures of Emil Gumbel on the statistics of extreme values, an area he helped found along with R. A. Fisher and Leonard Tippett. Gumbel was not only a mathematician, but also a prominent anti-Nazi activist. In the 1920s, he documented political murders by the Sturmabteilung, the Nazi paramilitary wing, and was one of the 33 signatories of the Dringender Appell, calling on non-Nazi politicans to unite “in the rejection of fascism” prior to the 1932 Weimar elections. They did not unite, and the Nazis went on to consolidate their grip on Weimar politics. Gumbel was forced to leave his Professorship at Heidelberg shortly afterwards, moving to France and then the US, where he taught at Columbia until his death in 1966.
Gumbel’s lectures were given, I guess, to the Applied Mathematics Division at the National Bureau of Standards. The basic idea is simple: if I take $n$ samples of a random variable with distribution $\mathcal{D}$, what does the maximum look like? From a practical perspective, outliers are sometimes more important than the bulk of observations, for instance with extreme weather events or fluctations of the financial market. If your goal is flood management, you don’t care about average rainfall!
This is related to a statistical error Gumbel calls the “three $\sigma$ fallacy” (and I suspect, the main reason the Bureau called him in). We are used to thinking that observations are “very likely” to fall within some number of standard deviations of the mean, say three for the normal distribution. But if I make enough observations, I will almost certainly get extreme outliers! If there is some small probability $p$ of being outside $3\sigma$, then with $n \approx 1/p$ observations, I would expect to get such an outlier.
My goal is to give a quick outline of these beautiful ideas, more or less as working notes for myself. I’ll start with some basic results, expanding on the $n \approx 1/p$ observation, and compute (asymptotically) the expected largest sample for a Gaussian. I’ll then talk about an analogue of the central limit theorem for maxima (due variously to Fisher, Tippett and Gnedenko) and end with some real-world applications.