On the penultimate tail behavior of Weibull-type models

Abstract

The Gumbel max-domain of attraction corresponds to a null tail index which do not distinguish the different tail weights that might exist between distributions within this class. The Weibull-type distributions form an important subgroup of this latter and includes the so-called \emph{Weibull-tail coefficient}, usually denoted θ, that specifies the tail behavior, with larger values indicating slower tail decay. Here we shall see that the Weibull-type distributions present a penultimate tail behavior Fréchet if θ>1 and a penultimate tail behavior Weibull whenever θ<1.