Volume 13, Issue 1

This article proposes a regression tree procedure to estimate conditional copulas. The associated algorithm determines classes of observations based on covariate values and fits a simple parametric copula model on each class. The association parameter changes from one class to another, allowing for non-linearity in the dependence structure modeling. It also allows the definition of classes of observations on which the so-called “simplifying assumption” holds reasonably well. When considering observations belonging to a given class separately, the association parameter no longer depends on the covariates according to our model. In this article, we derive asymptotic consistency results for the regression tree procedure and show that the proposed pruning methodology, i.e., the model selection techniques selecting the appropriate number of classes, is optimal in some sense. Simulations provide finite sample results, and an analysis of data of cases of human influenza presents the practical behavior of the procedure.

Kendall’s tau and conditional Kendall’s tau matrices are multivariate (conditional) dependence measures between the components of a random vector. For large dimensions, available estimators are computationally expensive and can be improved by averaging. Under structural assumptions on the underlying Kendall’s tau and conditional Kendall’s tau matrices, we introduce new estimators that have a significantly reduced computational cost while keeping a similar error level. In the unconditional setting, we assume that, up to reordering, the underlying Kendall’s tau matrix is block structured with constant values in each of the off-diagonal blocks. Consequences on the underlying correlation matrix are then discussed. The estimators take advantage of this block structure by averaging over (part of) the pairwise estimates in each of the off-diagonal blocks. Derived explicit variance expressions show their improved efficiency. In the conditional setting, the conditional Kendall’s tau matrix is assumed to have a block structure, for some value of the conditioning variable. Conditional Kendall’s tau matrix estimators are constructed similarly as in the unconditional case by averaging over (part of) the pairwise conditional Kendall’s tau estimators. We establish their joint asymptotic normality and show that the asymptotic variance is reduced compared to the naive estimators. Then, we perform a simulation study that displays the improved performance of both the unconditional and conditional estimators. Finally, the estimators are used for estimating the value at risk of a large stock portfolio; backtesting illustrates the obtained improvements compared to the previous estimators.

Due to their simple analytic form (bivariate) Archimedean copulas are usually viewed as very smooth and handy objects, which should distribute mass in a fairly regular and certainly not in a pathological way. Building upon recently established results on the Archimedean family and working with iterated function systems with probabilities, we falsify this natural conjecture and derive the surprising result that for every s ∈ s\hspace{0.33em}\in [1, 2] there exists some bivariate Archimedean copula A s {A}_{s} fulfilling that the Hausdorff dimension of the support of A s {A}_{s} is exactly s s .

This article investigates the effects of discrete marginal distributions on copula-based Markov chains. We establish results on mixing properties and parameter estimation for a copula-based Markov chain model with Bernoulli( p p ) marginal distributions, emphasizing some distinctions between continuous and discrete state-space Markov chains. We derive parameter estimators using the maximum-likelihood estimation (MLE) method and explore alternative estimators of p p that are asymptotically equivalent to the MLE. Furthermore, we provide the asymptotic distributions of these parameter estimators. A simulation study is conducted to evaluate the performance of the various estimators for p p . Additionally, we employ the likelihood ratio test to assess independence within the sequence.

This article introduces a hybrid framework that combines local Gaussian correlation (LGC) with hidden Markov models (HMMs) to model dynamic and nonlinear dependencies in general insurance claims, thereby addressing the limitations of static copula methods. When applied to Kenyan motor insurance claims (2008–2021) and Norwegian home insurance data (2012–2018), the proposed LGC-HMM approach captures regime-specific, nonlinear dependency patterns, revealing distinct stable and crisis periods through structural breaks in the dependency structure. Diagnostic checks confirm the HMM’s ability to reduce residual serial dependence, validating the latent state dynamics. Regime-aware value-at-risk (VaR) and tail VaR estimates derived from the LGC-HMM, using a proposed simulation procedure, outperform static copula models by adapting to structural changes, demonstrating robust forecasting performance. Visualization of forecasts via LGC maps further illustrates evolving tail dependencies. These findings support improved risk diversification and crisis-sensitive pricing strategies in actuarial practice.

This note is concerned with some historical remarks on and a partial review of two interesting mathematical subjects, the generalized Hoeffding-Fréchet functionals and the Monge-Kantorovich mass transportation problem. Both topics have a different motivation and history and are often considered in the literature as different subjects. The main aim of this review is to point out the close connection of these topics and to indicate some possibly fruitful relationships. For the class of Hoeffding-Fréchet functionals risk bounds for a lot of well-motivated additional model constraints have been worked out in recent years. These kinds of constraints are motivated by risk applications. We indicate some interesting connections of these developments to mass transportation problems as, e.g., to the solution of nonlinear mass transportation problems or to the use of stochastic ordering methods to the solution of mass transportation problems. We also briefly indicate the development of algorithms in the transportation problem by the regularization method (entropic optimal transport) and give corresponding references in the literature.