gumbel_v

Fisher-Tippett theorem with an historical perspective

A couple of weeks ago, Rafael asked me if I had something on the history of extreme value theory. Since I will get back to fundamental results about extremes in my course, I promised I will write down a short post on all that.
To start from the beginning, in 1928, Ronald Fisher and Leonard Tippett formulated the three types of limiting distributions for the maximum term of a random sample (Fisher & Tippett (1928)). The problem was to characterize function http://freakonometrics.blog.free.fr/public/perso5/ext-01.gif such that

http://freakonometrics.blog.free.fr/public/perso5/ext-2.gif

where http://freakonometrics.blog.free.fr/public/perso5/ext-3.gif where http://freakonometrics.blog.free.fr/public/perso5/ext-4.gif's are i.i.d. with cumulative distribution function http://freakonometrics.blog.free.fr/public/perso5/ext-5.gif. They had supporting arguments, but no (rigorous) proof. Nevertheless, the obtained that the only possible types for G were

http://freakonometrics.blog.free.fr/public/perso5/ext-6.gif

i.e. Fréchet type (Pareto-type tails), or

http://freakonometrics.blog.free.fr/public/perso5/ext-7.gif

i.e. Weibull type (bounded distribution type), or

http://freakonometrics.blog.free.fr/public/perso5/ext-8.gif

i.e. Gumbel type (exponential-type tails). Emil Gumbel has been intensively using the so-called Gumbel distribution on river flows, since (as he explained in 1958), "it seems that the rivers know the theory. It only remains to convince the engineers of the validity of this analysis".
Independently of that work (published in 1928), Maurice Fréchet considered in 1927 (in Sur la loi de probabilité de l'écart maximum) possible limits of

http://freakonometrics.blog.free.fr/public/perso5/ext-9.gif

and obtained only http://freakonometrics.blog.free.fr/public/perso5/ext-10.gif as possible limit. Richard von Mises gave in 1936 sufficient, but not necessary conditions for their (max) domain of attraction, i.e. characterization of function http://freakonometrics.blog.free.fr/public/perso5/ext-11.gif such that the maxima converges to some specific function http://freakonometrics.blog.free.fr/public/perso5/ext-01.gif (von Mises (1936)). E.g. he noticed that a sufficient condition on http://freakonometrics.blog.free.fr/public/perso5/ext-11.gifto be in the (max) domain of attraction of the Gumbel distribution is that

http://freakonometrics.blog.free.fr/public/perso5/ext-13.gif

Then in 1943, Boris Gnedenko gave a complete characterization of those three types, with a complete characterization for two of them (heavy tails, i.e. Fréchet type and bounded support, i.e. Weibull) but his necessary and sufficient condition was based on a function that was not explicitly defined (see Gnedenko (1943)). Laurens de Haan in the 70's derived checkable condition for Gumbel's type.
Boris Gnedenko proved (in Section 4 of his paper) that F is the (max) domain of attraction of http://freakonometrics.blog.free.fr/public/perso5/ext-10.gif if and only if http://freakonometrics.blog.free.fr/public/perso5/ext-16.gif is regularly varying at infinity, with index http://freakonometrics.blog.free.fr/public/perso5/ext-17.gif (even if the term "regular variation" was not mentioned in the paper). Similar results were derived to characterize functions in the (max) domain of attraction of Weibull. For the (max) domain of attraction of http://freakonometrics.blog.free.fr/public/perso5/ext-18.gif, Boris Gnedenko obtained that a necessary and sufficient condition was that there exists a function http://freakonometrics.blog.free.fr/public/perso5/ext-19.gif such http://freakonometrics.blog.free.fr/public/perso5/ext-19.gif goes to 0 at infinity and

http://freakonometrics.blog.free.fr/public/perso5/ext-20.gif

Several papers have discussed what function http://freakonometrics.blog.free.fr/public/perso5/ext-19.gif could be e.g. David Mejzler in 1949 (in Russian, but see also his 1965 paper), and Laurens de Hann in 1970 and 1971 (following the dramatic flood in the Netherlands in 1953, researchers in the Netherlands have focuses on dikes, and extreme value applications).

Mejzler's idea was to work on quantiles, and not on the cumulative distribution function. I.e. define

http://freakonometrics.blog.free.fr/public/perso5/ext-21.gif

Then a necessary and sufficient condition for F to be in the (max) domain of attraction of http://freakonometrics.blog.free.fr/public/perso5/ext-18.gif is that

http://freakonometrics.blog.free.fr/public/perso5/ext-23.gif

Laurens de Haan proved in 1971 that function http://freakonometrics.blog.free.fr/public/perso5/ext-19.gif can be - in general - given by

http://freakonometrics.blog.free.fr/public/perso5/ext-25.gif

And in 1976, Laurens de Haan obtained a three-type convergence working on quantile function http://freakonometrics.blog.free.fr/public/perso5/ext-26.gif (with a much shorter proof).
There have been many many papers extending Fisher-Tippett's theorem, e.g. on non-independent sequences, like exchangeable ones (in a paper by Simeon Berman in 1962, or on stationary Gaussian sequences in 1964).


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One thought on “Fisher-Tippett theorem with an historical perspective”

  1. Before seeing the reference.credit at the bottom I was going to comment on how great it was to find a piece from Robert Crumb in a math post!!!

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