Stability postulate

In probability theory, to obtain a nondegenerate limiting distribution for extremes of samples, 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 ${\displaystyle \ \mathbb {P} \left(X_{j}=x\right)\equiv f_{X}(x)\ ,}$

then the cumulative distribution function ${\displaystyle \ F_{Y_{n}}\ }$ for ${\displaystyle \ Y_{n}\equiv \max\{\ X_{1},\ \ldots ,\ X_{n}\ \}\ }$ is given by the simple relation

${\displaystyle F_{Y_{n}}(y)=\left[\ F_{X}(y)\ \right]^{n}~.}$

If there is a limiting distribution for the distribution of interest, the stability postulate states that the limiting distribution must be for some sequence of transformed or "reduced" values, such as ${\displaystyle \ \left(\ a_{n}\ Y_{n}+b_{n}\ \right)\ ,}$ where ${\displaystyle \ a_{n},\ b_{n}\ }$ may depend on  n  but not on x . This equation was obtained by Maurice René Fréchet and also by Ronald Fisher.

Only three possible distributions

To distinguish the limiting cumulative distribution function from the "reduced" greatest value from ${\displaystyle \ F(x)\ ,}$ we will denote it by ${\displaystyle \ G(y)~.}$ It follows that ${\displaystyle \ G(y)\ }$ must satisfy the functional equation

${\displaystyle \ \left[\ G\!\left(y\right)\ \right]^{n}=G\!\left(\ a_{n}\ y+b_{n}\ \right)~.}$

Boris Vladimirovich Gnedenko has shown there are no other distributions satisfying the stability postulate other than the following three: ^[1]

• Gumbel distribution for the minimum stability postulate
  • If ${\displaystyle \ X_{i}={\textrm {Gumbel}}\left(\ \mu ,\ \beta \right)\ }$ and ${\displaystyle \ Y\equiv \min\{\ X_{1},\ \ldots ,\ X_{n}\ \}\ }$ then ${\displaystyle \ Y\sim a_{n}\ X+b_{n}\ ,}$
    where ${\displaystyle \ a_{n}=1\ }$ and ${\displaystyle \ b_{n}=\beta \ \log n\$ ;}
  • In other words, ${\displaystyle \ Y\sim {\textsf {Gumbel}}\left(\ \mu -\beta \ \log n\ ,\ \beta \ \right)~.}$

• Weibull distribution (extreme value) for the maximum stability postulate
  • If ${\displaystyle \ X_{i}={\textsf {Weibull}}\left(\ \mu ,\ \sigma \ \right)\ }$ and ${\displaystyle \ Y\equiv \max\{\,X_{1},\ldots ,X_{n}\,\}\ }$ then ${\displaystyle \ Y\sim a_{n}\ X+b_{n}\ ,}$
    where ${\displaystyle \ a_{n}=1\ }$ and ${\displaystyle \ b_{n}=\sigma \ \log \!\left({\tfrac {1}{n}}\right)\$ ;}
  • In other words, ${\displaystyle \ Y\sim {\textsf {Weibull}}\left(\ \mu -\sigma \log \!\left({\tfrac {1}{n}}\ \right)\ ,\ \sigma \ \right)~.}$

• Fréchet distribution for the maximum stability postulate
  • If ${\displaystyle \ X_{i}={\textsf {Frechet}}\left(\ \alpha ,\ s,\ m\ \right)\ }$ and ${\displaystyle \ Y\equiv \max\{\ X_{1},\ \ldots ,\ X_{n}\ \}\ }$ 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, ${\displaystyle \ Y\sim {\textsf {Frechet}}\left(\ \alpha ,n^{\tfrac {1}{\alpha }}s\ ,\ m\ \right)~.}$

References

1. Gnedenko, B. (1943). "Sur La Distribution Limite Du Terme Maximum D'Une Serie Aleatoire". Annals of Mathematics. 44 (3): 423–453. doi:10.2307/1968974.