Generalized extreme value distribution について

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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 as type I, II and III extreme value distributions. By the extreme value theorem the GEV distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. Note that a limit distribution need not exist: this requires regularity conditions on the tail of the distribution. Despite this, the GEV distribution is often used as an approximation to model the maxima of long (finite) sequences of random variables.
In some fields of application the generalized extreme value distribution is known as the Fisher–Tippett distribution, named after Ronald Fisher and L. H. C. Tippett who recognised three function forms outlined below. However usage of this name is sometimes restricted to mean the special case of the Gumbel distribution.
==Specification==
The generalized extreme value distribution has cumulative distribution function
:

F(x;\mu,\sigma,\xi) = \exp\left\\right)\right" TITLE="1+\xi\left(\frac\right)\right">)^\right\}

for

1+\xi(x-\mu)/\sigma>0

, where

\mu\in\mathbb R

is the location parameter,

\sigma>0

the scale parameter and

\xi\in\mathbb R

the shape parameter. Thus for

\xi>0

, the expression just given for the cumulative distribution function is valid for

x > \mu-\sigma/\xi

, while for

\xi<0

it is valid for

x < \mu+ \sigma/(-\xi)

. In the first case, at the lower end-point it equals 0; in the second case, at the upper end-point, it equals 1. For

\xi = 0

the expression just given for the cumulative distribution function is formally undefined and is replaced by the result obtained by taking the limit as

\xi\to 0

F(x;\mu,\sigma,0) = \exp\left\\right)\right\}

,
without any restriction on ''x''.
The density function is, consequently,
:

f(x;\mu,\sigma,\xi) = \frac\left()^

\exp\left\\right)\right" TITLE="1+\xi\left(\frac\right)\right">)^\right\}

again, for

x > \mu-\sigma/\xi

in the case

\xi>0

, and for

x < \mu+\sigma/(-\xi)

in the case

\xi<0

. The density is zero outside of the relevant range. In the case

\xi=0

the density is positive on the whole real line and equal to
:

f(x;\mu,\sigma,\xi) = \frac\exp\left()

\exp\left\\right)\right" TITLE="\left(-\frac\right)\right">)\right\}

.
Example of density functions for distributions of the GEV family.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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