A society is said to exhibit more cumulative deprivation (affluence) when more individuals score low (high) in all dimensions of well-being. To measure these concepts, we adopt a position-based approach which builds on copulas.

First, we summarize the information about cumulative deprivation using the cdf of the maximal position across all dimensions, which is determined by the diagonal section of the copula. Second, we measure cumulative affluence through the cdf of the minimal position, which is closely related to the diagonal section of the survival copula.

We depict these functions into a single hairpin-type diagram along with two benchmarks: independence and perfect alignment. From this diagram, we define an index of cumulative deprivation (affluence) as the normalized area between the diagonal section of the copula (survival copula) and that representing independence. We discuss the properties of these indices and show its relationship with multivariate Spearman’s footrule. Several examples with well-known multivariate copulas illustrate our results.

We derive nonparametric estimators of the indices above and we apply them to analyse how cumulated deprivation and affluence have evolved worldwide from 2007 to 2021. The application involves yearly data on the three dimensions included in the Human Development Index: health, education and income.

Motivated by a recently established result saying that within the family of bivariate Archimedean copulas standard pointwise convergence implies the generally stronger weak conditional convergence,
i.e. convergence of almost all conditional distributions, this result is extended to the class of multivariate Archimedean copulas.
Working with the fact that generators of Archimedean copulas are $d$-monotone functions, pointwise convergence within the family of multivariate Archimedean copulas is characterized in terms of convergence of the corresponding generators,
derivatives of the generators, marginal copulas as well as marginal densities.
Furthermore, weak conditional convergence is a consequence of any of the afore-mentioned properties.
Utilizing that generators of Archimedean copulas can be represented via Williamson transforms of one dimensional probability measures,
it is established that weak convergence of the probability measures is equivalent to uniform convergence of the Archimedean copulas.
Using Markov kernels, Archimedean copulas inherit absolute continuity, singularity and discreteness from the afore-mentioned probability measures,
leading to the surprising result that absolutely continuous, singular, as well as discrete copulas are dense in the class of Archimedean copulas with respect to the uniform metric.

In the talk we consider strategies for model selection of copulas within parametric families and for preventing overfitting. The parameter is estimated by an approximate minimizer of the estimated Cramér-von Mises divergence supplemented by a regularization term. We provide results on strong consistency and on asymptotic normality of the estimator. Moreover, we consider statistical tests in order to compare the approximation quality of several model classes.