Revisiting Relations between Stochastic Ageing and Dependence for Exchangeable Lifetimes with an Extension for the IFRA/DFRA Property

We first review an approach that had been developed in the past years to introduce concepts of "bivariate ageing" for exchangeable lifetimes and to analyze mutual relations among stochastic dependence, univariate ageing, and bivariate ageing. A specific feature of such an approach dwells on the concept of semi-copula and in the extension, from copulas to semi-copulas, of properties of stochastic dependence. In this perspective, we aim to discuss some intricate aspects of conceptual character and to provide the readers with pertinent remarks from a Bayesian Statistics standpoint. In particular we will discuss the role of extensions of dependence properties. "Archimedean" models have an important role in the present framework. In the second part of the paper, the definitions of Kendall distribution and of Kendall equivalence classes will be extended to semi-copulas and related properties will be analyzed. On such a basis, we will consider the notion of "Pseudo-Archimedean" models and extend to them the analysis of the relations between the ageing notions of IFRA/DFRA-type and the dependence concepts of PKD/NKD

An overview of the goodness-of-fit test problem for copulas

We review the main "omnibus procedures" for goodness-of-fit testing for copulas: tests based on the empirical copula process, on probability integral transformations, on Kendall's dependence function, etc, and some corresponding reductions of dimension techniques. The problems of finding asymptotic distribution-free test statistics and the calculation of reliable p-values are discussed. Some particular cases, like convenient tests for time-dependent copulas, for Archimedean or extreme-value copulas, etc, are dealt with. Finally, the practical performances of the proposed approaches are briefly summarized

Convergence of Archimedean Copulas

Convergence of a sequence of bivariate Archimedean copulas to another Archimedean copula or to the comonotone copula is shown to be equivalent with convergence of the corresponding sequence of Kendall distribution functions.No extra differentiability conditions on the generators are needed.Archimedean copula;generator;Kendall distribution function

The importance of being the upper bound in the bivariate family

Any bivariate cdf is bounded by the Fréchet-Hoeffding lower and upper bounds. We illustrate the importance of the upper bound in several ways. Any bivariate distribution can be written in terms of this bound, which is implicit in logit analysis and the Lorenz curve, and can be used in goodness-of-fit assesment. Any random variable can be expanded in terms of some functions related to this bound. The Bayes approach in comparing two proportions can be presented as the problem of choosing a parametric prior distribution which puts mass on the null hypothesis. Accepting this hypothesis is equivalent to reaching the upper bound. We also present some parametric families making emphasis on this bound

The importance of being the upper bound in the bivariate family.

Any bivariate cdf is bounded by the Fr ´echet-Hoeffding lower and upper bounds. We illustrate the importance of the upper bound in several ways. Any bivariate distribution can be written in terms of this bound, which is implicit in logit analysis and the Lorenz curve, and can be used in goodness-of-fit assesment. Any random variable can be expanded in terms of some functions related to this bound. The Bayes approach in comparing two proportions can be presented as the problem of choosing a parametric prior distribution which puts mass on the null hypothesis. Accepting this hypothesis is equivalent to reaching the upper bound. We also present some parametric families making emphasis on this bound

Convergence of Archimedean Copulas

Convergence of a sequence of bivariate Archimedean copulas to another Archimedean copula or to the comonotone copula is shown to be equivalent with convergence of the corresponding sequence of Kendall distribution functions.No extra differentiability conditions on the generators are needed.

Estimating the risk-adjusted capital is an affair in the tails

(Re)insurance companies need to model their liabilities' portfolio to compute the risk-adjusted capital (RAC) needed to support their business. The RAC depends on both the distribution and the dependence functions that are applied among the risks in a portfolio. We investigate the impact of those assumptions on an important concept for (re)insurance industries: the diversification gain. Several copulas are considered in order to focus on the role of dependencies. To be consistent with the frameworks of both Solvency II and the Swiss Solvency Test, we deal with two risk measures: the Value-at-Risk and the expected shortfall. We highlight the behavior of different capital allocation principles according to the dependence assumptions and the choice of the risk measure.Capital Allocation, Copula, Dependence, Diversification Gain, Model Uncertainty, Monte Carlo Methods, Risk-Adjusted Capital, Risk Measure