Improved Semi-Parametric Bounds for Tail Probability and Expected Loss: Theory and Applications

2024-04-03

AI Summary

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.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Many management decisions involve accumulated random realizations for which only the first and second moments of their distribution are available. The sharp Chebyshev-type bound for the tail probability and Scarf bound for the expected loss are widely used in this setting. We revisit the tail behavior of such quantities with a focus on independence. Conventional primal-dual approaches from optimization are ineffective in this setting. Instead, we use probabilistic inequalities to derive new bounds and offer new insights. For non-identical distributions attaining the tail probability bounds, we show that the extreme values are equidistant regardless of the distributional differences. For the bound on the expected loss, we show that the impact of each random variable on the expected sum can be isolated using an extension of the Korkine identity. We illustrate how these new results open up abundant practical applications, including improved pricing of product bundles, more precise option pricing, more efficient insurance design, and better inventory management. For example, we establish a new solution to the optimal bundling problem, yielding a 17% uplift in per-bundle profits, and a new solution to the inventory problem, yielding a 5.6% cost reduction for a model with 20 retailers.

Problem

• Deriving new bounds for tail probability and expected loss using probabilistic inequalities
• Analyzing extreme values and their equidistant property in non-identical distributions
• Applying new bounds to practical problems like pricing, insurance, and inventory management

Innovation

• Uses probabilistic inequalities for new bounds
• Extends Korkine identity for expected loss
• Applies to product bundling and inventory management