Stability postulate

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 ${\displaystyle X_{1},X_{2},\dots ,X_{n}}$ are independent random variables with common probability density function

${\displaystyle p_{X_{j}}(x)=f(x),}$

then the cumulative distribution function of ${\displaystyle X'_{n}=\max\{\,X_{1},\ldots ,X_{n}\,\}}$ is

${\displaystyle F_{X'_{n}}={[F(x)]}^{n}}$

If there is a limiting distribution of interest, the stability postulate states that the limiting distribution is some sequence of transformed "reduced" values, such as ${\displaystyle (a_{n}X'_{n}+b_{n})}$, where ${\displaystyle a_{n},b_{n}}$ 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

${\displaystyle {[G(x)]}^{n}=G{(a_{n}x+b_{n})}}$

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:^[1]

• Gumbel distribution for the minimum stability postulate
  • If ${\displaystyle X_{i}={\textrm {Gumbel}}(\mu ,\beta )}$ and ${\displaystyle Y=\min\{\,X_{1},\ldots ,X_{n}\,\}}$ then ${\displaystyle Y\sim a_{n}X+b_{n}}$ where ${\displaystyle a_{n}=1}$ and ${\displaystyle b_{n}=\beta \log(n)}$
  • In other words, ${\displaystyle Y\sim {\textrm {Gumbel}}(\mu -\beta \log(n),\beta )}$
• Extreme value distribution for the maximum stability postulate
  • If ${\displaystyle X_{i}={\textrm {EV}}(\mu ,\sigma )}$ and ${\displaystyle Y=\max\{\,X_{1},\ldots ,X_{n}\,\}}$ then ${\displaystyle Y\sim a_{n}X+b_{n}}$ where ${\displaystyle a_{n}=1}$ and ${\displaystyle b_{n}=\sigma \log({\tfrac {1}{n}})}$
  • In other words, ${\displaystyle Y\sim {\textrm {EV}}(\mu -\sigma \log({\tfrac {1}{n}}),\sigma )}$
• Fréchet distribution for the maximum stability postulate
  • If ${\displaystyle X_{i}={\textrm {Frechet}}(\alpha ,s,m)}$ and ${\displaystyle Y=\max\{\,X_{1},\ldots ,X_{n}\,\}}$ then ${\displaystyle Y\sim a_{n}X+b_{n}}$ where ${\displaystyle a_{n}=n^{-{\tfrac {1}{\alpha }}}}$ and ${\displaystyle b_{n}=m\left(1-n^{-{\tfrac {1}{\alpha }}}\right)}$
  • In other words, ${\displaystyle Y\sim {\textrm {Frechet}}(\alpha ,n^{\tfrac {1}{\alpha }}s,m)}$