Many multivariate data sets exhibit a form of positive dependence, which can either appear globally between all variables or only locally within particular subgroups. A popular notion of positive dependence that allows for localized positivity is positive association. In this work we introduce a notion of positive dependence based on positive association for the parametric family of Hüsler–Reiss distributions, which can be considered as an analogue of multivariate Gaussians in multivariate extremes. Due to its parametric description we call this new notion of extremal positive dependence the metric property. As the parameter of a Hüsler–Reiss distribution is a Euclidean distance matrix, we observe that the metric property relates to research in electrical network theory and Euclidean geometry. We show that the metric property can be localized for Hüsler–Reiss graphical models and study surrogate likelihood inference for this setting. This gives rise to a two-step estimation procedure for locally metrical Hüsler–Reiss graphical models. The second step allows for a simple dual problem, which is implemented via a gradient descent algorithm. Finally, we demonstrate our results on simulated and real data.
This study develops a concentration inequality for random variables that satisfy Bernstein’s condition when an appropriate cover is present. This inequality generalizes standard concentration results for common distributions, including Gamma, Exponential, and bounded random variables. We extend this analysis to financial risk management by deriving an upper bound for the Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) of dependent portfolios with an independent subgroup structure. Numerical simulations demonstrate that the derived upper bounds for the VaR and TVaR of a dependent portfolio are tight.
We study conditional distributions and their quantiles when a continuous covariate is conditioned on an interval event, without discretizing the covariate itself, focusing on bivariate copulas. For strict bivariate Archimedean copulas, we show that interval-conditioned quantiles on an arbitrary bin can be characterized by a one-dimensional equation involving only the copula generator. We identify settings in which numerical integration is unnecessary and provide closed-form expressions for interval-conditioned quantiles in the first (threshold) bin for the Clayton, Frank, Gumbel, and Joe copulas. For bivariate Clayton and Frank copulas, we further establish a unique anchor value inside the bin at which the interval-conditioned quantile equals the pointwise conditional quantile. We further show that last bin formulas can be obtained via the survival copula and derive closed-form last bin quantiles and anchors for the bivariate Frank copula. Numerical illustrations show that equal-probability bins can make distinct copula families appear nearly indistinguishable at central quantiles, whereas tail-favored bins restore separation. Our results offer practical formulas for interval-based inference, which is relevant to privacy-motivated binning.
Extending bivariate dependence concepts to higher dimensions is a challenging but essential task for a comprehensive understanding of multivariate dependence. Moreover, measuring overall dependence based on averages across the full domain of the joint distribution may fail to discern changes in dependence across different segments of the distribution, especially in the tails. In order to incorporate these features, we present the multivariate tail concentration function (TCF) as a graphical tool to assess both global and tail dependence. We show that this tool allows to represent multivariate dependence in a 2D plot regardless of the number of dimensions, it quantifies both lower and upper tail dependence at a finite scale, and it relates to multivariate Blomqvist’s beta. We propose to estimate the TCF non-parametrically using two methods and we compare their finite sample performance through a simulation study. To illustrate its practical application, we use the TCF to evaluate co-movements among the six Spanish banks included in the IBEX 35 stock index.
Bivariate time-to-event data often arise in various fields, including medicine, engineering, and economics, where understanding the association between two survival times is crucial. Traditional global association measures like Spearman’s rho and Kendall’s tau provide an average assessment, but fail to capture how association evolves over time. Local association measures, on the other hand, including the so-called cross ratio function (CRF), have been proposed to look at the association in more detail. This paper introduces a novel nonparametric estimator for the CRF applicable for univariate right-censored data, relying on Bernstein polynomials to obtain a smooth estimate of the bivariate survival copula, its partial derivatives, and the copula density. The proposed estimator’s finite-sample performance is evaluated through an elaborate simulation study and applied to real-life data, highlighting its practical utility and setting the stage for future research on local association in survival analysis.
