Title: On left truncation invariant limit distributions under low threshold
Authors: Claudio Ignazzi - Università del Salento, Lecce, Italy (Italy) [presenting]
Fabrizio Durante - University of Salento (Italy)
Piotr Jaworski - University of Warsaw (Poland)
Abstract: Given a random pair $(X, Y)$ distributed according to $C(F, F)$, where $C$ is a bivariate copula and $F$ is a continuous univariate distribution function, the limit distribution of $(X, Y)$ given that the values of $X$ fall under a low threshold is studied. This limit distribution is defined in three different ways depending on which of the three classes of univariate marginals includes $F$, based on its rate of decay at minus infinity: Frechet, Weibull or Gumbel. Various assumptions on the copula $C$ are needed, such as exchangeability as well as information on its tail behavior, such as having a non-zero lower tail dependence coefficient. After computing the above limit distributions in the three cases considered, the (unique) copula of the three limit distributions is determined. It turns out to be the same copula for all three cases. Finally, the above copula is proved to be invariant under univariate truncation of the first variable, or left truncation invariant.