en>Tony1 at 13:18, 16 April 2013

2013-04-16T13:18:03Z

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{{unreferenced|date=March 2011}}
{{Expert-subject|statistics|date=May 2011}}

In [[probability theory]], to obtain a nondegenerate limiting distribution of the [[extreme value distribution]], it is necessary to "reduce" the actual greatest value by applying a [[linear transformation]] with coefficients that depend on the sample size.

If <math> X_1, X_2, \dots , X_n \, </math> are [[independence (probability theory)|independent]] [[random variable]]s with common probability density function

: <math> p_{X_j}(x)=f(x),</math>

then the [[cumulative distribution function]] of <math> X'_n=\max\{\,X_1,\ldots,X_n\,\} \, </math> is

: <math> F_{X'_n}={[F(x)]}^n \, </math>

If there is a limiting distribution of interest, the '''stability postulate''' states the limiting distribution is some sequence of transformed "reduced" values, such as <math>(a_n X'_n + b_n) \,</math>, where <math>a_n, b_n \,</math> may depend on ''n'' but not on ''x''.

To distinguish the limiting [[cumulative distribution function]] from the "reduced" greatest value from ''F''(''x''), we will denote it by ''G''(''x''). It follows that ''G''(''x'') must satisfy the [[functional equation]]

: <math> {[G(x)]}^n = G{(a_n x + b_n)} \, </math>

This equation was obtained by [[Maurice René Fréchet]] and also by [[Ronald Fisher]].

[[Boris Vladimirovich Gnedenko]] has shown there are ''no other'' distributions satisfying the stability postulate other than the following:

* [[Gumbel distribution]] for the ''minimum'' stability postulate
** If <math> X_i=\textrm{Gumbel}(\mu,\beta) \, </math> and <math> Y=\min\{\,X_1,\ldots,X_n\,\} \, </math> then <math> Y \sim a_n X+b_n \,</math> where <math>a_n=1\,</math> and <math> b_n= \beta \log(n) \, </math>
** In other words, <math> Y \sim \textrm{Gumbel}(\mu - \beta \log(n),\beta) \,</math>
* [[Extreme value distribution]] for the maximum stability postulate
** If <math> X_i=\textrm{EV}(\mu,\sigma) \, </math> and <math> Y=\max\{\,X_1,\ldots,X_n\,\} \, </math> then <math> Y \sim a_n X+b_n \,</math> where <math>a_n=1\,</math> and <math> b_n= \sigma \log(\tfrac{1}{n}) \, </math>
** In other words, <math> Y \sim \textrm{EV}(\mu - \sigma \log(\tfrac{1}{n}),\sigma) \,</math>
* [[Fréchet distribution]] for the maximum stability postulate
** If <math> X_i=\textrm{Frechet}(\alpha,s,m) \, </math> and <math> Y=\max\{\,X_1,\ldots,X_n\,\} \, </math> then <math> Y \sim a_n X+b_n \,</math> where <math>a_n=n^{-\tfrac{1}{\alpha}}\,</math> and <math> b_n= m \left( 1- n^{-\tfrac{1}{\alpha}}\right) \, </math>
** In other words, <math> Y \sim \textrm{Frechet}(\alpha,n^{\tfrac{1}{\alpha}} s,m) \,</math>

[[Category:Probability theory]]
[[Category:Extreme value data]]

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en>Tony1 at 13:18, 16 April 2013