On certain transformation of Archimedean copulas: Application to the non-parametric estimation of their generators

Elena Di Bernardino (1) , Didier Rullière (2)

1 CEDRIC - Centre d'études et de recherche en informatique et communications
2 SAF - Laboratoire de Sciences Actuarielle et Financière

We study the impact of certain transformations within the class of Archimedean copulas. We give some admissibility conditions for these transformations, and define some equivalence classes for both transformations and generators of Archimedean copulas. We extend the $r$-fold composition of the diagonal section of a copula, from $r \in \N$ to $r \in \R$. This extension, coupled with results on equivalence classes, gives us new expressions of transformations and generators. Estimators deriving directly from these expressions are proposed and their convergence is investigated. We provide confidence bands for the estimated generators. Numerical illustrations show the empirical performance of these estimators.

Mots clés: Transformations of Archimedean copula; self-nested diagonal; non-parametric estimation; tail dependence

Domaines: Probabilités [math.PR]; Méthodologie [stat.ME]; Economies et finances

Elena Di Bernardino, Didier Rullière. On certain transformation of Archimedean copulas: Application to the non-parametric estimation of their generators. Dependence Modeling, 2013, 1, pp.Pages 1-36, ISSN (Online) 2300-2298. ⟨10.2478/demo-2013-0001⟩. ⟨hal-00834000v3⟩