The value of shape constraints in discrete moment problems: a review and extension

This research reviews the use of shape constraints in discrete moment problems. In particular, we investigate the impact of incorporating logconcavity as the new shape constraint in two settings including bounding the k -out-of- n type probabilities and expectations of higher order convex functions of discrete random variables with non-negative and finite support. The bounds are obtained as the optimum values of non-convex nonlinear optimization problem that can be reformulated as a bilinear optimization problem. Numerical experiments show significant improvement in the tightness of the bounds when the shape of underlying unknown probability distribution is prescribed into moment bounding problems. Shape constraints also add values to calculation of the expected stop-loss of aggregated insurance claims within a fixed period. Our finding is expected to expand the scope of applications for both discrete moment problems and logconcavity.