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

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.

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