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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=1000000n=100M=matrix(runif(s),n,s/n)V=apply(M,2,max)bn=1an=1/nU=(V-bn)/anhist(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 , thenwhich means that the limiting distribution is Fréchet's.set.seed(1)s=1000000n=100M=matrix((runif(s))^(-1/2),n,s/n)V=apply(M,2,max)bn=0an=n^(1/2)U=(V-bn)/anhist(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 , theni.e. the limiting distribution for the maximum is Gumbel's distribution.library(evd)set.seed(1)s=1000000n=100M=matrix(rexp(s,1),n,s/n)V=apply(M,2,max)(bn=qexp(1-1/n))log(n)an=1U=(V-bn)/anhist(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 getas . 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=10000000n=1000M=matrix(rnorm(s,0,1),n,s/n)V=apply(M,2,max)(bn=qnorm(1-1/n,0,1))an=1/bnU=(V-bn)/anhist(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 , iftheni.e. using Taylor's approximation on the right term,This gives us normalizing coefficients we should use here.set.seed(1)s=10000000n=1000M=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)/anhist(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")