Robust Lambda-quantiles and extremal distributions

2024-06-19

AI Summary

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.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

In this paper, we investigate the robust models for $Lambda$-quantiles with partial information regarding the loss distribution, where $Lambda$-quantiles extend the classical quantiles by replacing the fixed probability level with a probability/loss function $Lambda$. We find that, under some assumptions, the robust $Lambda$-quantiles equal the $Lambda$-quantiles of the extremal distributions. This finding allows us to obtain the robust $Lambda$-quantiles by applying the results of robust quantiles in the literature. Our results are applied to uncertainty sets characterized by three different constraints respectively: moment constraints, probability distance constraints via the Wasserstein metric, and marginal constraints in risk aggregation. We obtain some explicit expressions for robust $Lambda$-quantiles by deriving the extremal distributions for each uncertainty set. These results are applied to optimal portfolio selection under model uncertainty.

Problem

• Extends classical quantiles to Λ-quantiles with probability/loss functions
• Derives robust Λ-quantiles using extremal distributions under partial information
• Applies results to portfolio selection with moment and Wasserstein constraints

Innovation

• Extends classical quantiles with Λ-probability/loss function
• Links robust Λ-quantiles to extremal distributions
• Applies to moment, Wasserstein, marginal constraints