distributionally robust tail bounding

This study addresses the challenge of accurately assessing tail risk under arbitrary non-decreasing aggregation functions when marginal distributions are fixed but the dependence structure is uncertain. To this end, it introduces a novel “hidden dependence” paradigm, constructing worst-case loss vectors driven by risk concentrations and joint tail events within a distributionally robust optimization framework. The analysis reveals how perturbations to a reference Gaussian dependence structure can significantly amplify tail risk. Theoretical results demonstrate that even infinitesimal deviations from the nominal dependence may lead to substantially higher capital requirements. Moreover, the work establishes worst-case risk bounds that coincide with those in the unconstrained setting and quantifies the extent of tail risk underestimation due to dependence uncertainty in credit risk applications.

This paper addresses the problem of estimating tail probabilities and expected shortfall for sums of independent random variables when only the first two moments are known—a setting where conventional semiparametric bounds are overly conservative. Methodologically, it introduces a novel analytical framework grounded in probabilistic inequalities: it identifies an equidistant distribution pattern of tail extrema, generalizes the Korkine identity to decouple individual variable contributions, and derives tighter tail bounds. Theoretically, this yields improved robust moment-based bounds. For the first time, the framework is applied to four operations and finance domains: bundle pricing, option pricing, insurance design, and inventory management. Empirical results demonstrate a 17% increase in per-bundle profit under optimal bundling and a 5.6% reduction in total cost in a 20-retailer inventory system. Collectively, the work significantly enhances the accuracy of robust decision-making under limited moment information in financial engineering and operations research.

This paper addresses robust Λ-quantile modeling under partial knowledge of the loss distribution, where the Λ-quantile generalizes the classical quantile via a flexible loss function Λ. To handle model uncertainty, we establish, for the first time, an equivalence between the robust Λ-quantile and the Λ-quantile under extremal distributions. Leveraging this equivalence, we derive closed-form analytical solutions for three canonical uncertainty sets: moment-constrained, Wasserstein-ball-constrained, and marginal-constrained sets. Our methodology integrates extremal distribution theory, optimal transport (Wasserstein distance), robust optimization, and risk aggregation modeling. The results unify the characterization of robustness across diverse uncertainty structures and yield computationally tractable, interpretable decision rules for optimal portfolio selection under model ambiguity.

This paper studies robust convex risk measures under distributional uncertainty, focusing on three canonical ambiguity sets: $p$-norm balls, Wasserstein balls, and mean-variance moment constraints. Leveraging convex analysis and duality theory, it derives— for the first time—the exact analytical forms of the corresponding convex conjugate penalty functions and obtains closed-form expressions for the robust risk measures over each ambiguity set. The main contributions are: (1) a unified analytical dual representation framework that characterizes the structural properties of worst-case risk; (2) simultaneous theoretical tractability and computational feasibility, substantially improving the efficiency of robust risk evaluation; and (3) a rigorous, implementable theoretical foundation for both Wasserstein-based robust optimization and moment-based robust models.

High-probability analysis of learning algorithms involving light-tailed (e.g., sub-exponential, sub-Gaussian) but possibly unbounded random variables poses significant technical challenges due to the lack of uniform concentration tools across distribution families. Method: We propose a generic black-box reduction that systematically transforms high-probability analysis of any algorithm relying on light-tailed randomness into the corresponding analysis under bounded-variable assumptions, incurring only controllable logarithmic-factor overheads. Contribution/Results: This is the first unified framework handling diverse light-tailed distributions without ad hoc concentration inequalities—greatly simplifying theoretical analysis. As applications, we reconstruct a generalized Azuma’s inequality and derive tight high-probability convergence bounds for stochastic optimization algorithms under light-tailed noise, demonstrating both the method’s effectiveness and broad applicability.

This work addresses the minimax estimation of discrete probability distributions under the $\ell_\infty$ norm, aiming to characterize optimal risk bounds both in expectation and with high probability. By integrating minimax theory, high-dimensional probability analysis, and empirical process techniques with constructive proofs, the study establishes the first fully computable, data-dependent tight risk bound, thereby resolving an open problem posed by Kontorovich and Painsky. The analysis also precisely identifies the structure of extremal distributions that achieve worst-case risk. These theoretical advances not only sharpen existing $\ell_\infty$ risk bounds but also lead to estimators that demonstrate superior empirical performance compared to current methods.

This paper addresses the problem of deriving sharp upper and lower bounds for distorted risk measures under distributional uncertainty. Methodologically, it introduces a generalized distorted risk analysis framework—free from continuity or monotonicity assumptions—that jointly incorporates mean, variance, unimodality, and Wasserstein distance constraints to characterize the ambiguity set; notably, it is the first to integrate unimodality with Wasserstein balls for extremal distribution derivation. Theoretically, it yields closed-form worst-case and best-case bounds for canonical distorted risk measures, including range Value-at-Risk and Gini deviation. Practically, the approach significantly enhances robustness against model misspecification and distributional shifts in portfolio optimization, delivering computationally tractable and interpretable guarantees for model risk assessment.

This work addresses the excessive conservatism of traditional worst-case risk measures under distributional uncertainty. The authors propose a weighted-average robust risk measurement framework that assigns higher weights to distributions closer to a reference model within an ambiguity set, replacing the worst-case evaluation with a weighted average of the underlying risk measure. Leveraging Banach lattice theory and Gelfand integration, they construct a convex risk measure exhibiting continuity and stability even for large ambiguity radii, and derive its dual representation. By integrating inf-convolution and quantile aggregation techniques, they establish the continuity, stability, and dominance properties of the proposed risk measure with respect to the ambiguity radius. Numerical experiments demonstrate the method’s superior calibration capability and sensitivity compared to conventional approaches.

This paper addresses robust risk aggregation and allocation under dependence uncertainty, focusing on the Range-Value-at-Risk (RVaR) and related variability measures within the Fréchet framework. Methodologically, it establishes the first sharp inequalities for RVaR, introduces extended convolution bounds, and integrates comonotonic analysis, nonconvex optimization, and dependence modeling. The theoretical contributions are threefold: (1) tight bounds for RVaR, inter-RVaR differences, and quantile differences; (2) characterization of optimal risk-sharing structures—comonotonic for large losses and countermonotonic for small losses/large gains; and (3) an explicit closed-form optimal allocation achieving the minimal average quantile-based risk, with a proof that no optimal solution exists when risks are unbounded above. These results strengthen the theoretical foundation of Fréchet problems in quantitative risk management.

This study addresses model uncertainty under non-convex and non-cash-additive risk measures by developing a robust quasiconvex risk measurement framework in general Lp spaces. By introducing an uncertainty set and integrating acceptance sets with capital allocation rules, the work employs functional analysis, duality theory, and c-quasiconvex analysis to propose two complementary mechanisms for generating robust risk measures. The main contribution lies in overcoming the classical limitations of convexity and cash additivity, establishing penalty-type dual representations for robust quasiconvex and cash-subadditive risk measures. Furthermore, the paper demonstrates that the structure of uncertainty itself can induce quasiconvexity, thereby revealing the fundamental impact of ambiguity on capital allocation and asset acceptability.