Last week, we had a discussion with some colleagues about the fact that – in order to prepare for the SOA exams – we did not have time (so far) to mention results on extreme values in our actuarial program. I did gave an introduction in my nonlife actuarial models class, but it was only an introduction, in three hours, in order to illustrate reinsurance pricing. And I told my students that if they wanted to know more about extreme values, they should start a master program in actuarial science and finance, since I will give a course on extremes (and copulas) next winter.
But actually, extreme values are everywhere ! For instance, there is a Prudential TV commercial where has people place large, round stickers on a number line to represent the age of the oldest person they know. This forms some kind of histogram. The message is to have Prudential prepare you to have adequate money for all these years. And actually, anyone can add his or her own sticker at the Prudential website.


Patrick Honner, on his blog (http://mrhonner.com/…), did mention this interesting representation. But this idea is not new, as mentioned in a post, published three years ago. In 1932, Emil Gumbel gave a talk in France on the “âge limite“. And as he wrote it “on peut donc supposer que la distribution de l’âge limite – c’est à dire la probabilité que cet âge ait une valeur donnée – soit Gaussienne“. In 1932 (not aware of Fisher and Tippett work, he thought that the limiting distribution for a maximum would be Gaussian). 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. And in 1937, he wrote a paper on “les centennaires” that can also be related to the work of Bortkiewicz on rare events. One should also mention one of the most important paper in extreme value theory, published in 1974 by Balkema and de Haan, on Residual Life Time at Great Age.
Because in this experiment, the question is “How Old is the Oldest Person You Know?“, so it is the distribution of a maximum. And from Fisher-Tippett theorem, if we assume that the age is bounded (and that there exists some finite upper limit), then the limiting distribution for the maxima (or to be more rigorous, a affine transformation of the maxima) should be Weibull distribution. And this is what it looks like
> plot(-x,dweibull(x,2.25,4),type="l",lwd=2)

As an actuary, the only thing I know about demography, is the distribution of the age of death. For instance, consider the following French life table
> alive <- read.table( + "https://perso.univ-rennes1.fr/arthur.charpentier/TV8890.csv", + sep=";",header=TRUE)$Lx > nb= -diff(alive) > ages=0:110 > plot(ages,nb,type="h")

This is the distribution of the age of the death in a given population. Which is not the same as the distribution mentioned above! What we look for is the following: given that someone is alive, what could be the distribution of his-her age ? Actually, if we assume that the yearly number of birth is constant with time (as well as death probability), then we can compute easily to number of people of age : we take everyone born (exactly)
years ago, and remove all those who died at at
,
, etc. So the function should be
> probadeath=nb/sum(nb) > nbx=function(x) 1-sum(probadeath[1:(x+1)]) > surv=Vectorize(nbx)(ages) > distrage=surv/sum(surv)
which looks like

But this assumption of constant number of birth is not that relevent. And actually, what we need is the distribution of the age within a population… This is a population pyramid, actually. The French one can be downloaded from http://www.insee.fr/fr/ppp/bases-de-donnees/….
> population <- read.table("popinsee2007.csv",sep=";",header=TRUE)$POPTOT07
> ages=0:107
> plot(ages,population/sum(population),type="h")

(the red line being the one obtained previously, using some natality assumptions). Now, let us use this population to generate acquaintances.
> agemax=function(nsim=1000,size=20){
+ agemax=rep(NA,nsim)
+ for(i in 1:nsim){
+ X=sample(ages,prob=population/sum(population),size=size,replace=TRUE)
+ agemax[i]=max(X)}
+ return(agemax)}
Here, we assume that everyone knows 20 other people, randomly chosen in the entire population, then we return the age of the oldest. And we do that for 1,000 people. Here is the distribution, we obtain
> XS=agemax(10000,20) > plot(table(XS)/length(XS),type="h",xlim=c(0,108))

where the red line is a Weibull distribution (a transformed one, actually, since in extremely value theory, the distance to the upper bound of the distribution has a Weibull density),
> library(MASS) > fit=fitdistr(108-XS,dweibull,list(shape=1,scale=1)) > lines(ages,dweibull(108-ages,fit$estimate[1],fit$estimate[2]),col="red")
Which is quite close to the distribution obtained in the commercial, don’t you think ? But still, it should be possible to be more accurate, since people should think of their parents, or grandparents. So I guess it could be possible to build a more accurate algorithm, to get something closer to the distribution obtained on the Prudential website. But first, let us wait to have more stickers, more observations… and then I’ll be back to play with it !









is the survival copula associated with
, in the sense that
















such that
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



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 
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 




(with a much shorter proof).
. 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
,

. Let
and
, then

. Let
and
, then

sample. We can use the following approximation of the cumulative distribution function (based on l’Hopital’s rule)
. Let
and
. Then we can get
. 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,

, if



stands for the cumulative distribution function, it should satisfy functional equation
, i.e. 
, i.e.
with
(i.e. finite lower bound for the support), i.e. 
if
(i.e. finite upper bound for the support), i.e. 

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.



, and below again,
}. When then look at the position of
and the one of the mean of 


, where
is the location parameter,
the scale parameter and
the shape parameter. Note that the expected value is

denotes the success probability. Then 
