On the penultimate tail behavior of Weibull-type models

TL;DR

This paper investigates the detailed tail behaviors of Weibull-type distributions, revealing a transition in penultimate tail behavior depending on the Weibull-tail coefficient, which refines understanding of their asymptotic properties.

Contribution

It characterizes the penultimate tail behavior of Weibull-type models, showing a phase transition based on the Weibull-tail coefficient , which was not previously detailed.

Findings

Weibull-type distributions exhibit Frchet penultimate tail behavior if >1.

They show Weibull penultimate tail behavior if <1.

The Weibull-tail coefficient  determines the tail decay rate.

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 \theta, 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 Fr\'echet if \theta>1 and a penultimate tail behavior Weibull whenever \theta<1.