Fisher–Tippett–Gnedenko 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),[1] Ronald Fisher and Leonard Henry Caleb Tippett (1928),[2] Mises (1936)[3][4] and Gnedenko (1943).[5]

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

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

or equivalently:

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

In such circumstances, the limit distribution ${\displaystyle G}$ belongs to either the Gumbel, the Fréchet or the Weibull family.[6]

In other words, if the limit above converges we will have ${\displaystyle G(x)}$ assume the form:[7]

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

for some parameters ${\displaystyle \gamma ,a,b}$. Remarkably, the right hand side is the cumulative distribution function of the generalized extreme value distribution (GEV) with extreme value index ${\displaystyle \gamma }$, scale parameter ${\displaystyle a}$ and location parameter ${\displaystyle b}$. 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 ${\displaystyle G(x)}$ above. The study of conditions for convergence of ${\displaystyle G}$ to particular cases of the generalized extreme value distribution began with Mises, R. (1936)[3][5][4] and was further developed by Gnedenko, B. V. (1943).[5]

Let ${\displaystyle F}$ be the distribution function of ${\displaystyle X}$, and ${\displaystyle X_{1},\dots ,X_{n}}$ an i.i.d. sample thereof. Also let ${\displaystyle x^{*}}$ be the populational maximum, i.e. ${\displaystyle x^{*}=\sup\{x\mid F(x)<1\}}$. The limiting distribution of the normalized sample maximum, given by ${\displaystyle G}$ above, will then be:[7]

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

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

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

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

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

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

See also

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

Notes

1. Fréchet, M. (1927), "Sur la loi de probabilité de l'écart maximum", Annales de la Société Polonaise de Mathématique, 6 (1): 93–116
2. Fisher, R.A.; Tippett, L.H.C. (1928), "Limiting forms of the frequency distribution of the largest and smallest member of a sample", Proc. Camb. Phil. Soc., 24 (2): 180–190, Bibcode:1928PCPS...24..180F, doi:10.1017/s0305004100015681
3. Mises, R. von (1936). "La distribution de la plus grande de n valeurs". Rev. Math. Union Interbalcanique 1: 141–160.
4. Falk, Michael; Marohn, Frank (1993). "Von Mises conditions revisited". The Annals of Probability: 1310–1328.
5. Gnedenko, B.V. (1943), "Sur la distribution limite du terme maximum d'une serie aleatoire", Annals of Mathematics, 44 (3): 423–453, doi:10.2307/1968974, JSTOR 1968974
6. Mood, A.M. (1950). "5. Order Statistics". Introduction to the theory of statistics. New York, NY, US: McGraw-Hill. pp. 251–270.
7. Haan, Laurens; Ferreira, Ana (2007). Extreme value theory: an introduction. Springer.