I will start here a short post on extreme values, with some historical perspective. In a recent paper (in French), I mentioned the use of the Pareto distribution as a standard model for extremes, but if reinsurers have been using the Pareto distribution for a long time (see here e.g.), the oldest mathematical models when dealing with extreme value should be related to work on maximum values in finite samples.
- The work of Ronald Fisher and Leonard Tippett
Leonard Henry Tippett, a former student of Karl Pearson published in Biometrika a note on extremes, in 1925. The goal was “the determination of the distribution of the range and the extremes for a large number of samples“. In 1925, everyone was looking for the Gaussian distribution everywhere, and Leonard Tippett observed that the distribution of the largest value did not have a Gaussian distribution.
A few years after, a joint work with Ronald Fisher was presented to the Cambridge Philosophical Society. The starting point was the idea of “stability” (even if the term did not appear explicitely in their work): the limiting distribution the maximum should be of the “same type” as the underlying distribution. Thus, if stands for the cumulative distribution function, it should satisfy functional equation
Solutions of that functional equation will give all possible limiting distributions. Thus, Fisher and Tippett obtained three possible limits,
- solutions of , i.e.
- solutions of , i.e. with (i.e. finite lower bound for the support), i.e.
- solutions of , i.e. if (i.e. finite upper bound for the support), i.e.
Based on those possible limiting distributions, Fisher and Tippett wanted to derive what has been called later on the “domain of attraction” of those distributions.
- The work of Maurice Fréchet, at the same time
In 1926, Maurice Fréchet wrote a paper on “la loi de probabilité de l’écart maximum“. That paper, as well as the one by Fisher and Tippett (wrote at the same time), investigated asymptotic limits. Both obtained functional equations, but only Maurice Fréchet understood the importance of the stability concept, pointed out by Paul Levy in the context of sums. Thus, Maurice Fréchet introduced the concept of what is called now “max-stability“. But Fréchet solve only functional equation . The point is that Fréchet studied absolute values of errors, i.e. strictly positive random variables. Thus, Maurice Fréchet considered distribution
where is an arbitrary positive constant. The “2” comes from the fact that Fréchet considered errors with respect to the median. But he did not introduced that new distribution function, he also proved that the distribution appears as a limit when the underlying distribution of the ‘s has an algebraic behavior at infinity, i.e. equivalent to , for some . I.e. he proved that Pareto-type tailed distibutions where in the domain of attraction of the Fréchet distribution.
- Later on, the work of Emil Gumbel
In 1932, Emil Gumbel gave a talk in France on the “âge limite“. But as he wrote it “on peut donc supposer que la distribution de l’âge limite – c’est à dire la probabilité que la probabilité de cet âge ait une valeur donnée – soit Gaussienne“. But a few years after, he read about Fisher’s work, and observed also that “la distribution d’une valeur extrêmes peut être représentée pour un nombre suffisant d’observations par la formule doublement exponentielle, pourvu que la distribution initiale se comporte asymptotiquement comme une exponentielle. La formule devient rigoureuse si la distribution initiale est exponentielle“, as he wrote in 1935. Thus, as Fréchet proved that Pareto type distribution were in the max-domain of attraction of Fréchet’s distribution, Gumbel obtained that exponential type distributions were in the max-domain of attraction of Gumbel’s distribution. He also introduced the term “distribution de type exponentiel“
For Emil Gumbel, it was natural to study the logarithmic derivative of the distribution, since it is the mortality rate in demography (area that Emil Gumbel studied previously). As he mentioned “d’un point de vue théorique, il est intéressant de noter que M. Fréchet a construit une distribution initiale d”une variable aléatoire pour laquelle la valeur absolue de la dérivée logarithmique diminue sans limite“. But since it was not a valuable property for practical applications, he decided that “nous nous bornerons au traitement des données de type exponentiel“. Emil Gumbel always tried to relate his work on extremes and what he did on demograpy.
For instance in 1937, he wrote a paper on “les centennaires” that can also be related to the work of Bortkiewicz on rare events. He also applied his work on radioactivity, and hydrology.
In the 30’s, hydrographs as Hazen or Graszberger introduced the concept of “yearly maximum” of
a river level. They actually proposed to look for actuarial models to study decennial or centennial floods. But they only used the lognormal distribution to model yearly maxima. In 1936, French hydrologist Aimé Coutagne met Emil Gumbel (who was teaching at the ISFA, in Lyon). At that time, Emil Gumbel was looking for possible applications (outside demography) for his doubly exponential distribution. As as pointed out by Aimé, “sa formule devait être applicable au cas des crues; c’est à dire des plus grands débits, problème analogue à celui des plus grands âges“. Not only Gumbel’s distribution gave better empirical results, but also it came with a theoritical justification.
- Gumbel’s distribution properties
Consider the Gumbel distribution, with location and scale parameters alpha and beta respectively, i.e.
Note that the associated quantile function is
- The work of Waloddi Weibull
Waloddi Weibull, a Swedish physict proposed a distribution in 1939, to represent the distribution of breaking strength of materials. He used it in the 50’s in reliability concept. Actually, Weibull appeared late in the story of extremes, since Fréchet, Fisher and Tippett mentioned it already in the mid-20’s.
- From the central limit theorem (on the average) to Fisher-Tippett theorem (on the maxima)
In order to visualize those two theorem, consider the following animation, where samples of 20 exponential variables are generated. From those 20 values, we plot the maximum in blue, and the average in red, on top. Just below, be rescale those points by considering , and below again, }. When then look at the position of and the one of the mean of . We then build an histogram to visualize the distribution of the rescaled maximum (in blue) and the rescale average (in red).
For those who might be busy, after 1000 generations of samples, we obtain the following histograms (below), including the Gaussian distribution below (i.e. the average of exponential variables looks Gaussian, even with only 20 observations, actually the Gaussian distribution is only asymptotic, i.e. we should consider samples of size 2000), and the maximum over 20 observations of exponential variables (on top) looks like a Gumbel distribution (actually, here it is the exact distribution, and it is the asymptotic distribution for exponential type variables).
- The GEV distribution
The unified expression of those three distributions is call the GEV distribution. The generalized extreme value distribution has cumulative distribution function
for , where is the location parameter, the scale parameter and the shape parameter. Note that the expected value is