MAT8886 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.
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.
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.
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,
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
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")
Credit: illustration is from Maurice Sendak's popular book where the wild things are, translated in French as Max et les Maximonstres.