Fisher–Tippett–Gnedenko theorem

This article is about the extreme value theorem in statistics. For the result in calculus, see extreme value theorem.

In statistics, the Fisher–Tippett–Gnedenko theorem (also the Fisher–Tippett theorem or the extreme value theorem) is a general result in extreme value theory regarding asymptotic distribution of extreme order statistics. The maximum of a sample of iid random variables after proper renormalization can only converge in distribution to one of 3 possible distributions, the Gumbel distribution, the Fréchet distribution, or the Weibull distribution. Credit for the extreme value theorem and its convergence details are given to Fréchet (1927), Ronald Fisher and Leonard Henry Caleb Tippett (1928), Mises (1936) and Gnedenko (1943).

The role of the extremal types theorem for maxima is similar to that of central limit theorem for averages, except that the central limit theorem applies to the average of a sample from any distribution with finite variance, while the Fisher–Tippet–Gnedenko theorem only states that if the distribution of a normalized maximum converges, then the limit has to be one of a particular class of distributions. It does not state that the distribution of the normalized maximum does converge.

Statement

Let $X_{1},X_{2},\ldots ,X_{n}$ be a sequence of independent and identically-distributed random variables with cumulative distribution function $F$. Suppose that there exist two sequences of real numbers $a_{n}>0$ and $b_{n}\in \mathbb {R}$ such that the following limits converge to a non-degenerate distribution function:

$\lim _{n\to \infty }P\left({\frac {\max\{X_{1},\dots ,X_{n}\}-b_{n}}{a_{n}}}\leq x\right)=G(x)$,

or equivalently:

$\lim _{n\to \infty }F^{n}\left(a_{n}x+b_{n}\right)=G(x)$.

In such circumstances, the limit distribution $G$ belongs to either the Gumbel, the Fréchet or the Weibull family.

In other words, if the limit above converges we will have $G(x)$ assume the form:

$G_{\gamma ,a,b}\left(x\right)=\exp \left(-(1+\gamma \,x_{a,b})^{-1/\gamma }\right),\;\;x_{a,b}={\frac {x-b}{a}},\;\;1+\gamma \,x_{a,b}>0$

for some parameters $\gamma ,a,b$. Remarkably, the right hand side is the cumulative distribution function of the generalized extreme value distribution (GEV) with extreme value index $\gamma$, scale parameter $a$ and location parameter $b$. The GEV distribution groups the Gumbel, Fréchet and Weibull distributions into a single one.

Conditions of convergence

The Fisher–Tippett–Gnedenko theorem is a statement about the convergence of the limiting distribution $G(x)$ above. The study of conditions for convergence of $G$ to particular cases of the generalized extreme value distribution began with Mises, R. (1936) and was further developed by Gnedenko, B. V. (1943).

Let $F$ be the distribution function of $X$, and $X_{1},\dots ,X_{n}$ an i.i.d. sample thereof. Also let $x^{*}$ be the populational maximum, i.e. $x^{*}=\sup\{x\mid F(x)<1\}$. The limiting distribution of the normalized sample maximum, given by $G$ above, will then be:

• A Fréchet distribution ($\gamma >0$) if and only if $x^{*}=\infty$ and $\lim _{t\rightarrow \infty }{\frac {1-F(tx)}{1-F(t)}}=x^{-1/\gamma }$ for all $x>0$.

In this case, possible sequences that will satisfy the theorem conditions are $b_{n}=0$ and $a_{n}=F^{-1}\left(1-{\frac {1}{n}}\right)$.

• A Weibull distribution ($\gamma <0$) if and only if $x^{*}$ is finite and $\lim _{t\rightarrow 0^{+}}{\frac {1-F(x^{*}-tx)}{1-F(x^{*}-t)}}=x^{-1/\gamma }$ for all $x>0$.

Possible sequences here are $b_{n}=x^{*}$ and $a_{n}=x^{*}-F^{-1}\left(1-{\frac {1}{n}}\right)$.

• A Gumbel distribution ($\gamma =0$) if and only if $\lim _{t\rightarrow 0^{-}}{\frac {1-F(t+xf(t))}{1-F(t)}}=e^{-x}$ with $f(t):={\frac {\int _{t}^{x^{*}}1-F(s)ds}{1-F(t)}}$.

Possible sequences here are $b_{n}=F^{-1}\left(1-{\frac {1}{n}}\right)$ and $a_{n}=f\left(F^{-1}\left(1-{\frac {1}{n}}\right)\right)$.

See also

• Extreme value theory
• Gumbel distribution
• Generalized extreme value distribution
• Pickands–Balkema–de Haan theorem
• Generalized Pareto distribution
• Exponentiated generalized Pareto distribution