Improved Semi-Parametric Bounds for Tail Probability and Expected Loss: Theory and Applications (2404.02400v3)
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.
- Axaster, S. (2000). Inventory Control. Springer, London, United Kingdom.
- Azuma, K. (1967). Weighted sums of certain dependent random variables. Tohoku Mathematical Journal, 19(3):357–367.
- Bentkus, V. (2004). On Hoeffding’s inequalities. The Annals of Probability, 32(2):1650 – 1673.
- Bernshtein, S. (1946). Theory of probability. Gostekhizdat, Moscow.
- Bhargava, H. K. (2013). Mixed bundling of two independently valued goods. Management Science, 59(9):2170–2185.
- Billingsley, P. (1995). Probability and Measure. Wiley Series in Probability and Statistics. Wiley.
- Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. The Annals of Mathematical Statistics, 23(4):493 – 507.
- Option pricing: a simplified approach. Journal of Financial Economics, 7(3):229–263.
- de la Peña, V. H. (1999). A general class of exponential inequalities for martingales and ratios. The Annals of Probability, 27(1):537–564.
- Option bounds. Journal of Applied Probability, 41(A):145–156.
- Self-Normalized Processes: Limit Theory and Statistical Applications. Springer Berlin Heidelberg, Berlin, Heidelberg.
- Eckalbar, J. C. (2010). Closed-form solutions to bundling problems. Journal of Economics & Management Strategy, 19(2):513–544.
- Freedman, D. A. (1975). On Tail Probabilities for Martingales. The Annals of Probability, 3(1):100 – 118.
- Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association, 102(477):359–378.
- Gordon, L. (1994). A stochastic approach to the gamma function. The American Mathematical Monthly, 101(9):858–865.
- Revisiting semidefinite programming approaches to options pricing: Complexity and computational perspectives. INFORMS Journal on Computing, 35(2):335–349.
- Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58(301):13–30.
- Lo, A. W. (1987). Semi-parametric upper bounds for option prices and expected payoffs. Journal of Financial Economics, 19(2):373–387.
- Mallows, C. L. (1956). Generalizations of Tchebycheff’s inequalities. Journal of the Royal Statistical Society: Series B (Methodological), 18(2):139–168.
- Marinelli, C. (2024). On some semi-parametric estimates for european option prices. Journal of Applied Probability, page 1–11.
- Mattner, L. (2003). Mean absolute deviations of sample means and minimally concentrated binomials. The Annals of Probability, 31(2):914 – 925.
- McDiarmid, C. (1989). On the method of bounded differences. In Surveys in Combinatorics, London Mathematical Society Lectures Notes 141, pages 148–188. Cambridge University Press, Cambridge, UK.
- Classical and New Inequalities in Analysis. Kluwer Academic Publishers, Springer Dordrecht, Amsterdam.
- Pinelis, I. (1994). Optimum bounds for the distributions of martingales in banach spaces. The Annals of Probability, 22(4):1679 – 1706.
- Scarf, H. (1958). A min-max solution of an inventory problem. Studies in the Mathematical Theory of Inventory and Production, 10(2):201–209.
- Distribution-free confidence bounds for Pr(X1+X2+⋯+Xk<Z)Prsubscript𝑋1subscript𝑋2⋯subscript𝑋𝑘𝑍\Pr(X_{1}+X_{2}+\cdots+X_{k}<Z)roman_Pr ( italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + ⋯ + italic_X start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT < italic_Z ). Journal of the American Statistical Association, 80(389):227–230.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.
