We propose a simple empirical representation of expectations such that: For a number of samples above a certain threshold, drawn from any probability distribution with finite fourth-order statistic, the proposed estimator outperforms the empirical average when tested against the actual population, with respect to the quadratic loss. For datasets smaller than this threshold, the result still holds, but for a class of distributions determined by their first four statistics. Our approach leverages the duality between distributionally robust and risk-averse optimization.

翻译：我们提出了一种简单的经验期望表示，其特性如下：对于超过特定阈值的样本数量，这些样本取自任意具有有限四阶统计量的概率分布，当以二次损失为准则与实际总体进行比较时，所提出的估计量优于经验平均。对于小于该阈值的数据集，该结论仍然成立，但仅适用于由前四阶统计量所确定的分布类。我们的方法利用了分布鲁棒优化与风险规避优化之间的对偶性。