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 $\ \mathbb\left( X_j = x \right) \equiv f_X(x)\ ,$

then the cumulative distribution function $\ F_\$ for $\ Y_n \equiv \max\\$ is given by the simple relation

: F_(y) = \left F_X(y)\ \rightn ~.

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 $\ \left(\ a_n\ Y_n + b_n\ \right)\ ,$ where $\ a_n,\ b_n\$ may depend on but not on .
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)\ \rightn = 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:

* Gumbel distribution for the ''minimum'' stability postulate
** If $\ X_i = \textrm\left(\ \mu,\ \beta \right)\$ and $\ Y \equiv \min\\$ then $\ Y \sim a_n\ X + b_n\ ,$
where $\ a_n = 1\$ and $\ b_n = \beta\ \log n\ ;$
** In other words, $\ Y \sim \textsf\left(\ \mu - \beta\ \log n\ ,\ \beta\ \right) ~.$

* Weibull distribution (extreme value) for the maximum stability postulate
** If $\ X_i = \textsf\left(\ \mu,\ \sigma\ \right)\$ and $\ Y \equiv \max\\$ then $\ Y \sim a_n\ X + b_n\ ,$
where $\ a_n = 1\$ and $\ b_n = \sigma\ \log\!\left( \tfrac \right)\ ;$
** In other words, $\ Y \sim \textsf\left(\ \mu - \sigma \log\!\left(\tfrac\ \right)\ ,\ \sigma\ \right) ~.$

* Fréchet distribution for the maximum stability postulate
** If $\ X_i=\textsf\left(\ \alpha,\ s,\ m\ \right)\$ and $\ Y \equiv \max\\$ then $\ Y \sim a_n\ X + b_n\ ,$
where $\ a_n = n^\$ and $\ b_n = m \left( 1 - n^ \right)\ ;$
** In other words, $\ Y \sim \textsf\left(\ \alpha,n^ s\ ,\ m\ \right) ~.$

References

Stability (probability)
Extreme value data

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