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期望经验表示方法：对于从任意具有有限四阶统计量的概率分布中抽取的超过某一阈值的样本数量，所提出的估计量在实际总体检验中，相对于二次损失而言，优于经验均值。对于小于该阈值的数据集，该结果仍然成立，但仅适用于由其前四阶统计量确定的一类分布。我们的方法利用了分布鲁棒优化与风险厌恶优化之间的对偶性。