Equality of types for the distribution of the maximum for two values of \(n\) implies extreme value type (Q1424665)

The following characterization of extreme value distributions is shown. Theorem. Assume that \(F\) is twice differentiable on its support and belongs to the domain of max-attraction of an extreme value distribution \(G_\gamma(x)=\exp(-(1+\gamma x)^{-1/\gamma})\). If there exist \(\alpha\not=1\), \(\alpha>0\), such that \(F(x)=F^\alpha(ax+b)\), then there exist \(c,d\) such that \(F(cx+d)=G_\gamma(x)\). Hence, if \(\xi\) and \(\eta\) are i.i.d. with smooth CDF \(F\) and the CDF of \(\max(\xi,\eta)\) is of the same type as \(F\), then \(F\) is of extreme-value type.