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 $X_1,X_2,\dots ,X_n$ are independent random variables with common probability density function $\mathbb{P}(X_j=x)\equiv F_X(x),$

then the cumulative distribution function $F_{Y_n}$ for $Y_n\equiv max\{X_1,\dots ,X_n\}$ is given by the simple relation

$F_{Y_n}(y)={[F_X(y)]}^n.$

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 $(a_n{Y}_n+b_n),$ where $a_n,b_n$ may depend on Template:Mvar but not on Template:Mvar. 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 $F(x),$ we will denote it by $G(y).$ It follows that $G(y)$ must satisfy the functional equation

${\left[G\left(y\right)\right]}^n=G(a_{n}y+b_n).$

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

• Gumbel distribution for the minimum stability postulate
  • If $X_i=\text{Gumbel}(\mu ,\beta )$ and $Y\equiv min\{X_1,\dots ,X_n\}$ then $Y\sim a_{n}X+b_n,$
    where $a_n=1$ and $b_n=\beta \log n;$
  • In other words, $Y\sim \mathsf{\text{𝖦𝗎𝗆𝖻𝖾𝗅}}(\mu -\beta \log n,\beta ).$

• Weibull distribution (extreme value) for the maximum stability postulate
  • If $X_i=\mathsf{\text{𝖶𝖾𝗂𝖻𝗎𝗅𝗅}}(\mu ,\sigma )$ and $Y\equiv max\{X_1,\dots ,X_n\}$ then $Y\sim a_{n}X+b_n,$
    where $a_n=1$ and $b_n=\sigma \log \left(\frac{1}{n}\right);$
  • In other words, $Y\sim \mathsf{\text{𝖶𝖾𝗂𝖻𝗎𝗅𝗅}}(\mu -\sigma \log \left(\frac{1}{n}\right),\sigma ).$

• Fréchet distribution for the maximum stability postulate
  • If $X_i=\mathsf{\text{𝖥𝗋𝖾𝖼𝗁𝖾𝗍}}(\alpha ,s,m)$ and $Y\equiv max\{X_1,\dots ,X_n\}$ then $Y\sim a_{n}X+b_n,$
    where $a_n=n^{-\frac{1}{\alpha }}$ and $b_n=m(1-n^{-\frac{1}{\alpha }});$
  • In other words, $Y\sim \mathsf{\text{𝖥𝗋𝖾𝖼𝗁𝖾𝗍}}(\alpha ,n^{\frac{1}{\alpha }}s,m).$

References

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