tail-risk analytics

Quantifying tail risk (e.g., financial crashes, extreme weather) under serially dependent data remains challenging due to the inadequacy of standard independence-based tail modeling. Method: This paper proposes a Bayesian inference framework based on the Generalized Pareto Distribution (GPD) that jointly models static marginal and dynamic conditional tail behavior of threshold exceedances, integrating beta-mixture dependence structures, heteroscedastic regression, and empirical log-likelihood theory—thereby relaxing the restrictive independence assumption. Contribution/Results: It establishes, for the first time under serial dependence, asymptotically honest Bayesian credible regions for tail parameters and derives theoretical conditions for prior admissibility. Simulation studies demonstrate substantial improvements over classical methods across ARMA, GARCH, and Markov copula models. Empirical applications to U.S. interest rates and Swiss electricity demand confirm high accuracy, robustness, and practical applicability in real-world tail risk assessment.

This paper addresses the dynamic tail-risk forecasting of extreme negative returns (>5%) in India’s Nifty 50 index. We propose the first Bayesian generalized Pareto regression model, which specifies a log-linear relationship between the GPD scale parameter and multivariate volatility covariates—including Indian, U.S., and gold market volatilities—thereby overcoming the limitations of static GPD modeling. Innovatively embedding Bayesian regression within extreme value theory, we systematically compare Cauchy, Lasso, Ridge, and g-prior specifications, demonstrating that the Cauchy prior achieves optimal balance between predictive accuracy and model parsimony. Empirical results show substantial improvements in RMSE, AIC, and BIC, alongside enhanced forecasting performance for extreme losses. Crucially, the model identifies, for the first time, the global spillover effect of U.S. equity volatility and the safe-haven effect of gold volatility—providing both theoretical foundations and practical guidance for cross-border risk management.

To address unreliable extreme value index (EVI) inference caused by scarcity of tail data, this paper proposes a perturbation-based synthetic exceedance generation method: controlled noise is injected into exceedances above a high threshold, followed by generalized Pareto distribution (GPD) modeling and construction of a consistent pivotal statistic. Innovatively, the perturbation mechanism is integrated with differential privacy guarantees; when GPD approximation error is substantial, a refined perturbation strategy is further introduced to enhance robustness. Experiments demonstrate that the proposed method significantly outperforms existing EVI inference approaches in terms of confidence interval coverage, width control, and resilience to model misspecification. It establishes a novel paradigm for reliable extreme-value analysis under sparse tail regimes.

This study addresses the joint identifiability and elicitability of tail risk measures—including Value-at-Risk (VaR), Expected Shortfall (ES), and Range Value-at-Risk (RVaR)—along with their associated quantiles. We establish, for the first time, necessary and sufficient conditions for their joint identifiability and elicitability. Methodologically, we construct a novel class of weighted scoring functions that uniformly generalizes the Fissler–Ziegel scoring family, enabling elicitation of previously non-elicitable functionals such as tail expectations conditional on quantiles. Our approach integrates distributional generators, generalized method of moments estimation, and regression modeling. The results provide a rigorous statistical foundation for tail risk modeling, substantially simplifying regression fitting, model comparison, and backtesting procedures. By ensuring coherent and robust evaluation of tail risk, this work enhances both the theoretical soundness and practical applicability of financial risk measurement.

Existing discrete distributions—such as Poisson, negative binomial, and their zero-inflated variants—struggle to model heavy-tailed integer-valued data, while the discrete generalized Pareto distribution (DGPD) suffers from sensitivity to high threshold selection and reliance on asymptotic tail approximations. To address these limitations, this paper proposes three flexible extensions of the DGPD: a full-support variant, a zero-inflated full-support variant, and a low-threshold tail-focused variant. Built upon a generalized Pareto discretization framework, these models integrate zero-inflation mechanisms and parameter-tunable structures, thereby eliminating dependence on arbitrary threshold choices and asymptotic assumptions. Parameter estimation is performed via maximum likelihood, and extensive simulation studies alongside real-data experiments demonstrate substantial improvements in overall goodness-of-fit and tail characterization. Across three benchmark scenarios, the proposed models reduce average estimation error by 18%–32% relative to standard baselines, consistently outperforming existing approaches.

This study addresses the challenge of effectively integrating heterogeneous risks across multiple scenarios in financial markets by proposing a Weighted Generalized Risk Measure (WGRM) and its associated Weighted Risk Quadrangle (WRQ), thereby extending the generalized risk measure and risk quadrangle framework to a weighted setting for the first time. Theoretically, the work establishes analytical characterizations of WGRM under both discrete and continuous settings, proving that its structural properties remain invariant and revealing intrinsic connections among risk, deviation, regret, and error under weighting. Computationally, it leverages convex analysis, stochastic optimization, and linear programming reformulation techniques to transform complex risk optimization problems into tractable linear programs. Empirical results demonstrate that portfolios constructed using WGRM significantly improve risk-adjusted returns, enhance downside resilience, and mitigate losses caused by misjudgments in individual scenarios on NASDAQ 100 and S&P 500 constituents.

Existing extreme-value models struggle to flexibly characterize both marginal tail behavior and joint dependence structures of bivariate threshold exceedances under asymptotic independence. This work proposes a novel bivariate subasymptotic parametric model that, while ensuring convergence of the margins to the generalized Pareto distribution, naturally captures the evolution of extremal dependence with varying thresholds through its scale parameters. The model encompasses the standard multivariate generalized Pareto distribution as a limiting special case and accommodates a broad spectrum of tail dependence patterns. Inference is carried out via a likelihood-free neural Bayesian approach with tailored priors, enabling direct computation and interpretation of failure probabilities. Extensive simulations and an analysis of Belgian rainfall extremes demonstrate the model’s flexibility and the effectiveness of the proposed inference framework.

This study addresses the challenge of estimating shape parameters in generalized Pareto distributions (GPD) for extreme-value data exhibiting clustering structures. The authors propose a novel approach that integrates graph-fused Lasso regularization into GPD modeling, enabling simultaneous identification of groups with similar tail behaviors and estimation of their respective shape parameters. This is the first application of graph-fused Lasso to extreme-value analysis, effectively capturing both homogeneity within and heterogeneity across clusters in terms of extremal characteristics. Theoretical analysis establishes the asymptotic properties of the proposed estimator, demonstrating its superior variance performance compared to conventional cluster-wise independent estimation. Simulation studies confirm reduced estimation variance and enhanced stability. When applied to precipitation extremes from 996 monitoring stations across Japan, the method successfully identifies geographically coherent regions sharing similar extreme rainfall patterns.

This study addresses the bias inherent in tail index estimation for heavy-tailed distributions by proposing a novel estimator that integrates bias correction with empirical likelihood. The method uniquely combines bias correction techniques within an empirical likelihood framework to yield a more accurate and stable estimator, accompanied by rigorous asymptotic theory. Simulation experiments demonstrate that the proposed approach significantly outperforms existing methods in finite samples, while empirical analyses on real-world data further confirm its practical effectiveness and applicability.

This study addresses the challenge of generating extreme joint loss scenarios and characterizing conditional distributions in multivariate heavy-tailed risk factor systems. It proposes a Self-Similar Generative Estimation (SSGEN) framework that models extremal dependence via Pareto radial components, learning from intermediate exceedances to extrapolate reliably to rarer events. The key insight is that both the conditional stress distribution and the most likely stress configuration are governed by a common limiting tail law, enabling a generative approach that ensures convergence even when the target event is absent from observed samples. The method accurately recovers rare-event probabilities and scaling laws for conditional stress scenarios, delivering a data-driven inverse stress solution with theoretical guarantees on convergence rates.