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Pair-copula constructions are flexible dependence models that use bivariate copulas as building blocks. In this article, we extend them with generalized additive models to allow covariates effects. Borrowing ideas from a traditionally univariate context, we let each pair-copula parameter depend directly on the covariates in a parametric, semiparametric, or nonparametric way. We propose a sequential estimation method that we study by simulation, and apply it to investigate the time-varying dependence structure between the intraday returns on four major foreign exchange rates. An R package, scripts reproducing the results in this article, and additional simulation results are provided as supplementary material.

We investigate a new approach to estimating a regression function based on copulas. The main idea behind this approach is to write the regression function in terms of a copula and marginal distributions. Once the copula and the marginal distributions are estimated, we use the plug-in method to construct our new estimator. Because various methods are available in the literature for estimating both a copula and a distribution, this idea provides a rich and flexible family of regression estimators. We provide some asymptotic results related to this copula-based regression modeling when the copula is estimated via profile likelihood and the marginals are estimated nonparametrically. We also study the finite sample performance of the estimator and illustrate its usefulness by analyzing data from air pollution studies.

Despite many fuzzy time series forecasting (FTSF) models addressing complex temporal patterns and uncertainties in time series data, two limitations persist: they do not treat fuzzy and crisp time series as a unified whole for analyzing nonlinear relationships between different moments, and they fail to effectively capture how uncertainty in temporal patterns affects predictions. In this paper, we propose an FTSF model integrating Bayesian networks to overcome the limitations. Bayesian network (BN) structure learning is employed to extract fuzzy–crisp dependencies between historical fuzzified data and predicted crisp data alongside temporal crisp dependencies within crisp data. Integrating fuzzy logical relationship groups (FLRGs) and the two BNs representing the fuzzy–crisp and crisp relationships identifies temporal patterns efficiently. BN parameter learning models the occurrence uncertainties of dependencies through conditional probability distributions in BNs, while fuzzy empirical conditional probabilities quantify the occurrence uncertainties of the elements in FLRGs. The defuzzification stage infers the crisp predicted value using the fuzzy-empirical-probability weighted FLRGs and the two BN. We validate the forecasting performance of the proposed model on sixteen diverse time series. Experimental results demonstrate the competitive forecasting performance of the proposed model compared to state-of-the-art methods.

In many time-to-event studies, the event of interest is recurrent. Here, the data for each sample unit correspond to a series of gap times between the subsequent events. Given a limited follow-up period, the last gap time might be right-censored. In contrast to classical analysis, gap times and censoring times cannot be assumed independent, i.e., the sequential nature of the data induces dependent censoring. Also, the number of recurrences typically varies among sample units leading to unbalanced data. To model the association pattern between gap times, so far only parametric margins combined with the restrictive class of Archimedean copulas have been considered. Here, taking the specific data features into account, we extend existing work in several directions: we allow for nonparametric margins and consider the flexible class of D-vine copulas. A global and sequential (one- and two-stage) likelihood approach are suggested. We discuss the computational efficiency of each estimation strategy. Extensive simulations show good finite sample performance of the proposed methodology. It is used to analyze the association of recurrent asthma attacks in children. The analysis reveals that a D-vine copula detects relevant insights, on how dependence changes in strength and type over time.

This paper proposes a fuzzy copula-based optimization framework for modeling dependence structures and financial risk under parameter uncertainty. The parameters of selected copula families are represented as trapezoidal fuzzy numbers, and their α-cut intervals capture both the support and core ranges of plausible dependence values. This fuzzification transforms the estimation of copula parameters into a fuzzy optimization problem, enhancing robustness against sampling variability. The methodology is empirically applied to gold and oil futures (1 January 2015–1 January 2025), comparing symmetric copulas, i.e., Gaussian and Frank and asymmetric copulas, i.e., Clayton, Gumbel and Student-t. The results prove that the fuzzy copula framework provides richer insights than classical point estimation by explicitly expressing uncertainty in dependence measures (Kendall’s τ, Spearman’s ρ) and risk indicators (Value-at-Risk, Conditional Value-at-Risk). Rolling-window analyses reveal that fuzzy VaR and fuzzy CVaR effectively capture temporal dependence shifts and tail severity, with fuzzy CVaR consistently producing more conservative risk estimates. This study highlights the potential of fuzzy optimization and fuzzy dependence modeling as powerful tools for quantifying uncertainty and managing extreme co-movements in financial markets.

We characterize absolutely continuous symmetric copulas with square integrable densities in this paper. This characterization is used to create new copula families, that are perturbations of the independence copula. The full study of mixing properties of Markov chains generated by these copula families is conducted. An extension that includes the Farlie–Gumbel–Morgenstern family of copulas is proposed. We propose some examples of copulas that generate non-mixing Markov chains, but whose convex combinations generate ψ-mixing Markov chains. Some general results on ψ-mixing are given. The Spearman’s correlation ρS and Kendall’s τ are provided for the created copula families. Some general remarks are provided for ρS and τ. A central limit theorem is provided for parameter estimators in one example. A simulation study is conducted to support derived asymptotic distributions for some examples.

This study proposes a bivariate distribution with Exponentiated Gumbel (BEG) marginals to estimate return levels of annual maximum daily rainfall (AMDR) in Mexico. We analyze 181 gauging stations across two contrasting climates (Coahuila, Tabasco) and compare BEG against Generalized Extreme Value (GEV), Gumbel (G), and Exponentiated Gumbel (EG). Parameters are estimated by maximum likelihood. Model selection uses AICc (primary) and BIC (tie-breaker), both computed from the same maximized log-likelihood. On a per-station basis, BEG yields the lowest AICc for 70% of samples. Differences in return levels become more pronounced at high non-exceedance probabilities. Monte Carlo reliability checks show that BEG reduces bias and mean squared error (MSE) relative to univariate fits. Using L-moments to delineate homogeneous regions and fitting all BEG pairs confirms these results. A worked example (station 5001) shows that bootstrap 95% CIs for BEG are narrower than for EG, illustrating reduced marginal-quantile uncertainty under joint estimation. Together, BEG provides a robust, dependence-aware tool for regional frequency analysis of extreme rainfall.

Modeling dependence in high-dimensional systems has become an increasingly important topic. Most approaches rely on the assumption of a multivariate Gaussian distribution such as statistical models on directed acyclic graphs (DAGs). They are based on modeling conditional independencies and are scalable to high dimensions. In contrast, vine copula models accommodate more elaborate features like tail dependence and asymmetry, as well as independent modeling of the marginals. This flexibility comes however at the cost of exponentially increasing complexity for model selection and estimation. We show a novel connection between DAGs with limited number of parents and truncated vine copulas under sufficient conditions. This motivates a more general procedure exploiting the fast model selection and estimation of sparse DAGs while allowing for non-Gaussian dependence using vine copulas. By numerical examples in hundreds of dimensions, we demonstrate that our approach outperforms the standard method for vine structure selection. Supplementary material for this article is available online.

Investigating the dependence structures among the characteristics of the current unhealthy air pollution events is a valuable endeavor to understand the pollution behavior more clearly and determine the potential future risks. This study determined the characteristics of air pollution events based on their duration, severity, and intensity. It focused on modeling the dependence structures for all the possible pairs of characteristics, which were (duration, intensity), (severity, intensity), and (duration, severity), using various parametric copula models. The appropriate copula models for describing the behavior of the relationship pairs of the (duration, intensity), (severity, intensity), and (duration, severity) were found to be the Tawn type 1, 180°-rotated Tawn type 1, and Joe, respectively. This result showed that the dependence structures for the pairs were skewed and asymmetric. Therefore, the obtained copulas were appropriate models for such non-elliptical structures. These obtained models can be further extended in future work through the vine copula approach to provide a more comprehensive insight into the tri-variate relationship of the duration–intensity–severity characteristics.

We introduce a new family of bivariate exponential distributions based on the counter-monotonic shock model. This family of distribution is easy to simulate and includes the Fréchet lower bound, which allows to span all degrees of negative dependence. The construction and distributional properties of the proposed bivariate distribution are presented along with an estimation of the parameters involved in our model based on the method of moments. A simulation study is carried out to evaluate the performance of the suggested estimators. An extension to the general model describing both negative and positive dependence is sketched in the last section of the paper.