Extreme Value Statistics: Gumbel, Fréchet & Weibull

Fisher-Tippett-Gnedenko theorem: maxima of i.i.d. samples converge to one of three universality classes.

Gumbel (Type I)

Fréchet (Type II)

Weibull (Type III)

Shape ξ (GEV) 0.00

Scale σ 1.00

Block size n 50

The Generalized Extreme Value (GEV) distribution unifies all three types via shape ξ: ξ=0 → Gumbel (exponential tails, e.g. Gaussian maxima); ξ>0 → Fréchet (power-law tails, e.g. Pareto maxima); ξ<0 → Weibull (bounded tail, e.g. uniform maxima).