Conference Agenda

In many applications involving binary variables, only pairwise dependence measures, such as correlations, are available. However, for multi-way tables involving more than two variables, these quantities do not uniquely determine the joint distribution, but instead define a family of admissible distributions that share the same pairwise dependence while potentially differing in higher-order interactions. In this talk, we introduce a geometric framework to describe the entire feasible set of such joint distributions with uniform margins. We show that this admissible set forms a convex polytope, analyze its symmetry properties, and characterize its extreme rays. These extremal distributions provide fundamental insights into how higher-order dependence structures may vary while preserving the prescribed pairwise information. Unlike traditional methods for table generation, which return a single table, our framework makes it possible to explore and understand the full admissible space of dependence structures, enabling more flexible choices for modeling and simulation. We illustrate the usefulness of our theoretical results through examples and a real case study on rater agreement.

Characteristics of d-variate risks, such as downside risk measures of aggregate positions or optimal portfolio values, play a central role in financial and actuarial applications. This talk addresses the question of when such characteristics are robust to (small) misspecifications in the copula.

Measures of association based on ranks, such as Spearman’s footrule[1], play a central role in multivariate statistics due to their robustness and invariance properties. However, classical versions of these coefficients are often unable to capture directional dependence structures that arise in high-dimensional settings. Motivated by this limitation and by the newly defined coefficients described subsequently[2] [3], we introduce a novel family of directional Spearman’s footrule coefficients designed to quantify multivariate dependence along prescribed directions in the unit d-dimensional hypercube.

The proposed coefficients are formulated within the framework of copula theory, which allows for a clear separation between marginal behavior and the underlying dependence structure. Our construction extends the classical Spearman’s footrule by incorporating directional information, enabling the detection of dependence patterns that remain undetected by standard measures. We establish a general definition for arbitrary dimensions and directions and investigate its main theoretical properties. In particular, we analyze their behavior under independence and maximal positive dependence, their relation to stochastic orders, as well as their relationship with marginal distributions and lower-dimensional structures. These properties are shown to be consistent with those of the classical footrule coefficient.

To facilitate practical implementation, we also introduce nonparametric estimators based on ranks. These estimators are easy to compute and suitable for multivariate data. Their asymptotic behavior is discussed, highlighting consistency and stability properties analogous to those of existing rank-based dependence measures.

Several illustrative examples are provided to demonstrate the usefulness of the proposed coefficients. Explicit expressions are derived for well-known families of d-copulas, including the Farlie–Gumbel–Morgenstern and Cuadras–Augé, allowing for a detailed analysis of how directional dependence varies with model parameters. These examples show that the proposed coefficients are able to distinguish different directional dependence patterns even when classical global measures coincide.

Overall, this work provides a new tool for directional dependence analysis in multivariate settings, complementing existing rank-based measures and offering a finer understanding of complex dependence structures with applications in finance, reliability, and multivariate risk analysis.

[1] Spearman, C. (1906). ‘Footrule’ for measuring correlation. Brithis Journal of Psychology, 2, 89-108.

[2] Úbeda-Flores, M. (2004). Multivariate versions of Blomqvist’s beta and Spearman’s footrule. Ann. Inst. Statist. Math., 57(4), 781-788.

[3] Decancq, K., Pérez, A., Prieto-Alaiz, M. (2025). Multivariate Dependence Based on Diagonal Sections: Spearman’s Footrule and Related Measures. In: Steland, A., Rafajłowicz, E., Parolya, N. (eds) Stochastic Models, Statistics and Their Applications. SMSA 2024. Springer Proceedings in Mathematics & Statistics, vol 499. Springer, Cham.

This talk introduces a sophisticated GARCH-EVT-Copula framework designed
to enhance portfolio risk estimation for multi-asset portfolios. We specically
address the limitations of the traditional Markowitz mean-variance model, which
often fails to account for extreme market events and tail dependencies. Our
approach integrates a Generalized Autoregressive Conditional Heteroscedasticity
(GARCH) model to capture time-varying volatility with Extreme Value Theory
(EVT) for modeling heavy-tailed distributions.

A key innovation presented is the application of product copulas to model the
intricate, non-linear dependencies among four diverse nancial indices: the MSCI
World IMI, ICE BofAML Global Government Index, HFRX Global Hedge Fund
Index, and Swiss Re Global Unhedged CAT Bond Performance Index. Product
copulas prove particularly eective in capturing asymmetric dependencies and
non-normal characteristics, leading to signicantly more accurate Value-at-Risk
(VaR) estimations.

Our empirical analysis demonstrates the superior performance of the product
copula framework compared to traditional models, particularly regarding its resilience
during the 2008 nancial crisis and its ecacy in equal risk contribution
portfolios.