Discrete Moment Problems with Logconcave and Logconvex Distributions
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We introduce new shape constraints, logconcavity and logconvexity, to discrete moment problems for 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 and convex nonlinear optimization problems, where the non-convex problem is reformulated as a bilinear optimization problem. We present numerical experiments to show the improvement in the tightness of the bounds when the shape of underlying unknown probability distribution is prescribed into discrete moment problems. We apply our optimization based bounding methodology in an insurance problem to estimate the expected stop-loss of aggregated insurance claims within a fixed period. The proposed bounding methodology is expected to expand the scope of applications for both discrete moment problems and logconcavity and logconvexity.