Didier Rullière

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

Abstract 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 ∈ N to r ∈ 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.

On tail dependence coefficients of transformed multivariate Archimedean copulas

This paper presents the impact of a class of transformations of copulas in their upper and lower multivariate tail dependence coefficients. In particular we focus on multivariate Archimedean copulas. In the first part of this paper, we calculate multivariate transformed tail dependence coefficients when the generator of the considered transformed copula exhibits some regular variation properties, and we investigate the behaviour of these coefficients in cases that are close to tail independence. We obtain new results under specific conditions involving regularly varying hazard rates of components of the transformation. These results are also valid for non-transformed Archimedean copulas. In the second part we deal with transformations presented by Di Bernardino and Rulliere 20. We show the utility of using transformed Archimedean copulas, as they permit to build Archimedean generators exhibiting any chosen couple of lower and upper tail dependence coefficients. Finally, we detail the extreme behaviour of the transformed radial part of Archimedean copulas (using results in Larsson and Neslehova 52) and we explain possible applications with transformed Markov chains.

On the estimation of Pareto fronts from the point of view of copula theory

A probabilistic framework is applied to Pareto front estimation.A link between Pareto front and level lines of a multivariate distribution is given.Parametric approximations of Pareto front are provided for Archimedean copulas.Consequences and limits of the Archimedean hypothesis are discussed.The full estimation methodology is analyzed on several examples. Given a first set of observations from a design of experiments sampled randomly in the design space, the corresponding set of non-dominated points usually does not give a good approximation of the Pareto front. We propose here to study this problem from the point of view of multivariate analysis, introducing a probabilistic framework with the use of copulas. This approach enables the expression of level lines in the objective space, giving an estimation of the position of the Pareto front when the level tends to zero. In particular, when it is possible to use Archimedean copulas, analytical expressions for Pareto front estimators are available. Several case studies illustrate the interest of the approach, which can be used at the beginning of the optimization when sampling randomly in the design space.

On hyperbolic iterated distortions for the adjustment of survival functions

This paper presents a class of distortions of survival functions. Studied distortions are built in order to respect several properties that seem to us useful in actuarial science. We focus here on some particular hyperbolic distortions, which preserve analytic invertibility of the distorted survival function, and for which inverse distortions belong to the same hyperbolic class. We propose some specific parameterizations of these distortions which give a straightforward inversion, and discuss the importance of such an inversion in insurance and finance. We prove the convergence of composed distortions to any target law, and give initial values and a particular methodology for the parameters estimation. Numerical figures illustrate the adaptation of these distortions to several actuarial fields.

Impact of Dependence on Some Multivariate Risk Indicators

The minimization of some multivariate risk indicators may be used as an allocation method, as proposed in Cénac et al. (Stat Risk Model 29(1):47–71, 2012). The aim of capital allocation is to choose a point in a simplex, according to a given criterion. In Maume-Deschamps et al. (2015), it is proved that the proposed allocation technique satisfies a set of coherence axioms. In the present one, we study the properties and asymptotic behavior of the allocation for some distribution models. We also analyze the impact of the dependence structure on the allocation using some copulas.

Nested Kriging predictions for datasets with a large number of observations

This work falls within the context of predicting the value of a real function at some input locations given a limited number of observations of this function. The Kriging interpolation technique (or Gaussian process regression) is often considered to tackle such a problem, but the method suffers from its computational burden when the number of observation points is large. We introduce in this article nested Kriging predictors which are constructed by aggregating sub-models based on subsets of observation points. This approach is proven to have better theoretical properties than other aggregation methods that can be found in the literature. Contrarily to some other methods it can be shown that the proposed aggregation method is consistent. Finally, the practical interest of the proposed method is illustrated on simulated datasets and on an industrial test case with

Multivariate Extensions Of Expectiles Risk Measures

10^4

Valuation of portfolio loss derivatives in an infectious model

104 observations in a 6-dimensional space.