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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DOI：
10.1287/moor.1040.0137
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发表时间：
2005-08
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影响因子：
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通讯作者：
I. Popescu
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作者：
I. Popescu
我们提供了一个优化框架，用于计算具有特殊性质的分布函数期望的最优上界和下界，给定矩约束。Bertsimas和Popescu（概率论中的最优不等式：凸优化方法。SIAM J.优化2004. Forthcoming）已经展示了如何通过半定规划获得任意分布的最优矩不等式。如果底层分布具有额外的结构属性，包括对称性、单峰性、凸性或光滑性，则这些边界不是尖锐的。对于凸分布类，在某种意义上产生的一个适当的参数家庭，我们使用圆锥对偶，以显示如何最佳的时刻界可以有效地计算为半定程序。特别是，我们得到的对称和单峰分布的切比雪夫不等式的推广，并提供数值计算比较这些界限，高阶矩。我们也将这些结果推广到多元分布。
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.