A semidefinite programming approach to optimal-moment bounds for convex classes of distributions

A semidefinite programming approach to optimal-moment bounds for convex classes of distributions

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Article ID: iaor20061379
Country: United States
Volume: 30
Issue: 3
Start Page Number: 632
End Page Number: 657
Publication Date: Aug 2005
Journal: Mathematics of Operations Research
Authors:
Keywords: programming (semidefinite)
Abstract:

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 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.

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