Fisher-Tippet-Gnedenko theorem

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Theorem

In general, for iid $\set{X_{i}}_{i=1}^{n}$ with cdf F(x) we have $\mathbb{P}[\max_{i=1}^{n}\set{X_{i}}>x]=F^{n}(x).$ Let $x_{*}:=\sup\set{x: F(x)<1}$, then as $n\to \infty$ \begin{equation*} \max_{i=1,...,n}\set{X_{i}}\stackrel{prob}{\to} x_{*}. \end{equation*} We normalize it by constants $a_{n}>0,b_{n}$ s.t. $$ \frac{\max_{i=1,...,n}\set{X_{i}}-b_{n}}{a_{n}}\stackrel{dis}{\to} X_{*},$$ for some non-degenerate $X_{*}$ with distribution $G(x)$. We have the following three possible limits

  • Frechet: If $x>0$, $G(x)=exp\{-x^{-a}\}$ for some $a>0$.If $x\leq 0$, $G(x)=0$.
  • Gumbel: If $x\in \mathbb{R}$, $G(x)=exp\{-e^{-\frac{x-\mu}{\sigma}}\}$ for $\mu\in \mathbb{R}, \sigma>0$.
  • Weibull: If $x\geq 0$, $G(x)=1-exp\{-(\frac{x}{\lambda})^{k}\}$ for $\lambda,k>0$. If $x< 0$ , $G(x)=0$.


Fisher-Tippet-Gnedenko theorem

  • G(x) will be Frechet if and only if $F(x)<1$ and $\lim_{t\to +\infty}\frac{1-F(tx)}{1-F(t)}= x^{-\theta}$ for $x>0$ and $\theta>0$. Here we can set $b_{n}:=0$ and $a_{n}:=F^{-1}(1-\frac{1}{n})$.
  • G(x) will be Weibull if and only if $x_{*}:=\sup\{x:F(x)<1\}<\infty$ and $\lim_{t\to +\infty}\frac{1-F(x_{*}+tx)}{1-F(x_{*}-t)}= (-x)^{-\theta}$ for $x<0$ and $\theta>0$. Here we can set $b_{n}:=x_{*}$ and $a_{n}:=x_{*}-F^{-1}(1-\frac{1}{n})$.
  • G will be Gumbel if the density $f(x):=\frac{d}{dx}F(x)>0$ and it is differentiable in $(x_{1},x_{*})$ for some $x_{1}$, and $\lim_{x\to x_{*}}\frac{d}{dx}[\frac{1-F(x)}{f(x)}]=0$. Here we can set $b_{n}:=F^{-1}(1-\frac{1}{n})$ and $a_{n}:=\frac{1}{nf(b_{n})}$.


Proof

Proof under construction