# 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 issue.

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 such that

where where ‘s are i.i.d. with cumulative distribution function . They had supporting arguments, but no (rigorous) proof. Nevertheless, the obtained that the only possible types for G were

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

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

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

and obtained only 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 such that the maxima converges to some specific function (von Mises (1936)). E.g. he noticed that a sufficient condition on to be in the (max) domain of attraction of the Gumbel distribution is that

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 if and only if is regularly varying at infinity, with index (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 , Boris Gnedenko obtained that a necessary and sufficient condition was that there exists a function such goes to 0 at infinity and

Several papers have discussed what function 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

Then a necessary and sufficient condition for F to be in the (max) domain of attraction of is that

Laurens de Haan proved in 1971 that function can be – in general – given by

And in 1976, Laurens de Haan obtained a three-type convergence working on quantile function (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).

# Fisher-Tippett theorem and limiting distribution for the maximum

Tomorrow, we will discuss Fisher-Tippett theorem. The idea is that there are only three possible limiting distributions for normalized versions of the maxima of i.i.d. samples . For bounded distribution, consider e.g. the uniform distribution on the unit interval, i.e. on the unit interval. Let and . Then, for all and ,

i.e. the limiting distribution of the maximum is Weibull’s.

```set.seed(1)
s=1000000
n=100
M=matrix(runif(s),n,s/n)
V=apply(M,2,max)
bn=1
an=1/n
U=(V-bn)/an
hist(U,probability=TRUE,,col="light green",
xlim=c(-7,1),main="",breaks=seq(-20,10,by=.25))
u=seq(-10,0,by=.1)
v=exp(u)
lines(u,v,lwd=3,col="red")```

For heavy tailed distribution, or Pareto-type tails, consider Pareto samples, with distribution function . Let and , then

which means that the limiting distribution is Fréchet’s.

```set.seed(1)
s=1000000
n=100
M=matrix((runif(s))^(-1/2),n,s/n)
V=apply(M,2,max)
bn=0
an=n^(1/2)
U=(V-bn)/an
hist(U,probability=TRUE,col="light green",
xlim=c(0,7),main="",breaks=seq(0,max(U)+1,by=.25))
u=seq(0,10,by=.1)
v=dfrechet(u,shape=2)
lines(u,v,lwd=3,col="red")```

For light tailed distribution, or exponential tails, consider e.g. a sample of exponentially distribution variates, with common distribution function . Let and , then

i.e. the limiting distribution for the maximum is Gumbel’s distribution.

```library(evd)
set.seed(1)
s=1000000
n=100
M=matrix(rexp(s,1),n,s/n)
V=apply(M,2,max)
(bn=qexp(1-1/n))
log(n)
an=1
U=(V-bn)/an
hist(U,probability=TRUE,col="light green",
xlim=c(-2,7),ylim=c(0,.39),main="",breaks=seq(-5,15,by=.25))
u=seq(-5,15,by=.1)
v=dgumbel(u)
lines(u,v,lwd=3,col="red")```

Consider now a Gaussian sample. We can use the following approximation of the cumulative distribution function (based on l’Hopital’s rule)

as . Let and . Then we can get

as . I.e. the limiting distribution of the maximum of a Gaussian sample is Gumbel’s. But what we do not see here is that for a Gaussian sample, the convergence is extremely slow, i.e., with 100 observations, we are still far away from Gumbel distribution,

and it is only slightly better with 1,000 observations,

```set.seed(1)
s=10000000
n=1000
M=matrix(rnorm(s,0,1),n,s/n)
V=apply(M,2,max)
(bn=qnorm(1-1/n,0,1))
an=1/bn
U=(V-bn)/an
hist(U,probability=TRUE,col="light green",
xlim=c(-2,7),ylim=c(0,.39),main="",breaks=seq(-5,15,by=.25))
u=seq(-5,15,by=.1)
v=dgumbel(u)
lines(u,v,lwd=3,col="red")```

Even worst, consider lognormal observations. In that case, recall that if we consider (increasing) transformation of variates, we are in the same domain of attraction. Hence, since , if

then

i.e. using Taylor’s approximation on the right term,

This gives us normalizing coefficients we should use here.

```set.seed(1)
s=10000000
n=1000
M=matrix(rlnorm(s,0,1),n,s/n)
V=apply(M,2,max)
bn=exp(qnorm(1-1/n,0,1))
an=exp(qnorm(1-1/n,0,1))/(qnorm(1-1/n,0,1))
U=(V-bn)/an
hist(U,probability=TRUE,col="light green",
xlim=c(-2,7),ylim=c(0,.39),main="",breaks=seq(-5,40,by=.25))
u=seq(-5,15,by=.1)
v=dgumbel(u)
lines(u,v,lwd=3,col="red")```