A Semidefinite Programming Approach to Optimal-Moment Bounds for Convex Classes of Distributions

We provide an optimization framework for computing optimal upper and lower bounds on functional expectations of distributions with special properties, given moment constraints. Bertsimas and Popescu (Optimal inequalities in probability theory: a convex optimization approach. SIAM J. Optim. 2004. Forthcoming) have already shown how to obtain optimal moment inequalities for arbitrary distributions via semidefinite programming. These bounds are not sharp if the underlying distributions possess additional structural properties, including symmetry, unimodality, convexity, or smoothness. For convex distribution classes that are in some sense generated by an appropriate parametric family, we use conic duality to show how optimal moment bounds can be efficiently computed as semidefinite programs. In particular, we obtain generalizations of Chebyshev’s inequality for symmetric and unimodal distributions and provide numerical calculations to compare these bounds, given higher-order moments. We also extend these results for multivariate distributions.