Tight tail probability bounds for distribution-free decision making

TL;DR

This paper develops new tight bounds on tail probabilities for bounded distributions using mean and mean absolute deviation, with applications in decision-making models and robust optimization.

Contribution

It introduces exact semi-infinite linear program solutions for tail bounds under bounded support, mean, and MAD, advancing distribution-free probabilistic analysis.

Findings

Derived tight lower and upper tail bounds for bounded distributions.

Applied bounds to distribution-free analysis in newsvendor, pricing, and reinsurance models.

Enabled convex reformulations of ambiguous chance constraints in robust optimization.

Abstract

Chebyshev's inequality provides an upper bound on the tail probability of a random variable based on its mean and variance. While tight, the inequality has been criticized for only being attained by pathological distributions that abuse the unboundedness of the underlying support and are not considered realistic in many applications. We provide alternative tight lower and upper bounds on the tail probability given a bounded support, mean and mean absolute deviation of the random variable. We obtain these bounds as exact solutions to semi-infinite linear programs. We leverage the bounds for distribution-free analysis of the newsvendor model, monopolistic pricing, and stop-loss reinsurance. We also exploit the bounds for safe approximations of sums of correlated random variables, and to find convex reformulations of single and joint ambiguous chance constraints that are ubiquitous in distributionally robust optimization.