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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 Extreme value theory or extreme value analysis (EVA) is a branch of statistics dealing with the extreme deviations from the median of probability distributions. It seeks to assess, from a given ordered sample of a given random variable, the ...
regarding asymptotic distribution of extreme
order statistic In statistics, the ''k''th order statistic of a statistical sample is equal to its ''k''th-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference. Importa ...
s. The maximum of a sample of iid
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
s after proper renormalization can only converge in distribution to one of 3 possible distributions, the
Gumbel distribution In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. T ...
, the
Fréchet distribution The Fréchet distribution, also known as inverse Weibull distribution, is a special case of the generalized extreme value distribution. It has the cumulative distribution function :\Pr(X \le x)=e^ \text x>0. where ''α'' > 0 is a ...
, or the
Weibull distribution In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Maurice R ...
. Credit for the extreme value theorem and its convergence details are given to Fréchet (1927), Fisher and
Tippett Tippett is a surname. Notable people with the surname include: * Andre Tippett (born 1959), American Hall of Fame footballer *Clark Tippet (1954–1992), American dancer *Dave Tippett (born 1961), ice hockey coach * Keith Tippett (born 1947), Eng ...
(1928),
Mises Mises or von Mises may refer to: * Ludwig von Mises, an Austrian-American economist of the Austrian School, older brother of Richard von Mises ** Mises Institute, or the Ludwig von Mises Institute for Austrian Economics, named after Ludwig von ...
(1936) and
Gnedenko Boris Vladimirovich Gnedenko (russian: Бори́с Влади́мирович Гнеде́нко; January 1, 1912 – December 27, 1995) was a Soviet Ukrainian mathematician and a student of Andrey Kolmogorov. He was born in Simbirsk (now Ulyanov ...
(1943). The role of the extremal types theorem for maxima is similar to that of
central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables thems ...
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 In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usua ...
with
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
F. Suppose that there exist two sequences of real numbers a_n > 0 and b_n \in \mathbb such that the following limits converge to a non-
degenerate distribution In mathematics, a degenerate distribution is, according to some, a probability distribution in a space with support only on a manifold of lower dimension, and according to others a distribution with support only at a single point. By the latter ...
function: : \lim_P\left(\frac\leq x\right) = G(x), or equivalently: : \lim_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 Weibull is a Swedish locational surname. The Weibull family share the same roots as the Danish / Norwegian noble family of Falsenbr>They originated from and were named after the village of Weiböl in Widstedts parish, Jutland, but settled in Skå ...
family Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Idea ...
. In other words, if the limit above converges, G(x) will assume the form: :G_\gamma(x) = \exp\left(-(1 + \gamma \, x)^ \right), \;\; 1 + \gamma \, x_ > 0 or else :G_0(x) = \exp\left(-\exp(-x)\right) for some parameter \gamma. This is the cumulative distribution function of the
generalized extreme value distribution In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known ...
(GEV) with extreme value index \gamma. The GEV distribution groups the Gumbel, Fréchet and Weibull distributions into a single one. Note that the second formula (the Gumbel distribution) is the limit of the first as \gamma goes to zero.


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 (1936) and was further developed by Gnedenko (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\. The limiting distribution of the normalized sample maximum, given by G above, will then be: *A
Fréchet distribution The Fréchet distribution, also known as inverse Weibull distribution, is a special case of the generalized extreme value distribution. It has the cumulative distribution function :\Pr(X \le x)=e^ \text x>0. where ''α'' > 0 is a ...
(\gamma > 0) if and only if x^* = \infty and \lim_ \frac = u^ for all u> 0. :This corresponds to what is called a heavy tail. In this case, possible sequences that will satisfy the theorem conditions are b_n = 0 and a_n=F^\left(1-\frac\right). *A
Gumbel distribution In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. T ...
(\gamma = 0), with x^* finite or infinite, if and only if \lim_ \frac = e^ for all u>0 with f(t) := \frac. : Possible sequences here are b_n = F^\left(1-\frac\right) and a_n = f\left(F^\left(1-\frac\right)\right). *A
Weibull distribution In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Maurice R ...
(\gamma < 0) if and only if x^* is finite and \lim_ \frac = u^ for all u>0. : Possible sequences here are b_n = x^* and a_n=x^* - F^\left(1-\frac\right).


Examples


Fréchet distribution

For the
Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fu ...
:f(x)=(\pi^2+x^2)^ the cumulative distribution function is: :F(x)=1/2+\frac 1\pi\arctan(x/\pi) 1-F(x) is
asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
to 1/x, or :\ln F(x)\sim-1/x and we have :\ln F(x)^n=n\ln F(x)\sim-n/x. Thus we have :F(x)^n\approx\exp(-n/x) and letting u=x/n-1 (and skipping some explanation) :\lim_F(nu+n)^n =\exp(-(1+u)^)= G_1(u) for any u. The expected maximum value therefore goes up linearly with .


Gumbel distribution

Let us take the
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu i ...
with cumulative distribution function :F(x)=\frac 12\text(-x/\sqrt 2). We have :\ln F(x)\sim-\frac and :\ln F(x)^n=n\ln F(x)\sim-\frac. Thus we have :F(x)^n\approx\exp\left(-\frac\right). If we define c_n as the value that satisfies :\frac=1 then around x=c_n :\frac\approx\exp(c_n(c_n-x)). As increases, this becomes a good approximation for a wider and wider range of c_n(c_n-x) so letting u=c_n(c_n-x) we find that :\lim_F(u/c_n+c_n)^n =\exp(-\exp(-u))= G_0(u). Equivalently, :\lim_P\Bigl(\frac \leq u \Bigr) =\exp(-\exp(-u))= G_0(u). We can see that \ln c_n\sim(\ln\ln n)/2 and then :c_n\sim\sqrt so the maximum is expected to climb ever more slowly toward infinity.


Weibull distribution

We may take the simplest example, a uniform distribution between 0 and 1, with cumulative distribution function :F(x)=x from 0 to 1. Approaching 1 we have :\ln F(x)^n=n\ln F(x)\sim-n(1-x). Then :F(x)^n\approx\exp(nx-n). Letting u=1+n(1-x) we have :\lim_F(u/n+1)^n=\exp\left(-(1-u)\right)=G_(u). The expected maximum approaches 1 inversely proportionally to .


See also

*
Extreme value theory Extreme value theory or extreme value analysis (EVA) is a branch of statistics dealing with the extreme deviations from the median of probability distributions. It seeks to assess, from a given ordered sample of a given random variable, the ...
*
Gumbel distribution In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. T ...
*
Generalized extreme value distribution In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known ...
* Pickands–Balkema–de Haan theorem * Generalized Pareto distribution
Exponentiated generalized Pareto distribution


Notes

{{DEFAULTSORT:Fisher-Tippett-Gnedenko theorem Theorems in statistics Extreme value data Tails of probability distributions