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
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 ...
. Suppose that there exist two sequences of real numbers
and
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:
:
,
or equivalently:
:
.
In such circumstances, the limit distribution
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,
will assume the form:
:
or else
:
for some parameter
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
. 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
goes to zero.
Conditions of convergence
The Fisher–Tippett–Gnedenko theorem is a statement about the convergence of the limiting distribution
above. The study of conditions for convergence of
to particular cases of the generalized extreme value distribution began with Mises (1936)
and was further developed by Gnedenko (1943).
Let
be the distribution function of
, and
an i.i.d. sample thereof. Also let
be the populational maximum, i.e.
. The limiting distribution of the normalized sample maximum, given by
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 ...
(
) if and only if
and
for all
.
:This corresponds to what is called a
heavy tail. In this case, possible sequences that will satisfy the theorem conditions are
and
.
*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 ...
(
), with
finite or infinite, if and only if
for all
with
.
: Possible sequences here are
and
.
*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 ...
(
) if and only if
is finite and
for all
.
: Possible sequences here are
and
.
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 ...
:
the cumulative distribution function is:
:
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
or
:
and we have
:
Thus we have
:
and letting
(and skipping some explanation)
:
for any
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
:
We have
:
and
:
Thus we have
:
If we define
as the value that satisfies
:
then around
:
As increases, this becomes a good approximation for a wider and wider range of
so letting
we find that
:
Equivalently,
:
We can see that
and then
:
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
:
from 0 to 1.
Approaching 1 we have
:
Then
:
Letting
we have
:
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 distributionExponentiated generalized Pareto distribution
Notes
{{DEFAULTSORT:Fisher-Tippett-Gnedenko theorem
Theorems in statistics
Extreme value data
Tails of probability distributions