The entropic risk measure is widely used in high-stakes decision-making across economics, management science, finance, and safety-critical control systems because it captures tail risks associated with uncertain losses. However, when data are limited, the empirical entropic risk estimator, formed by replacing the expectation in the risk measure with a sample average, underestimates true risk. We show that this negative bias grows superlinearly with the standard deviation of the loss for distributions with unbounded right tails. We further demonstrate that several existing bias reduction techniques developed for empirical risk either continue to underestimate entropic risk or substantially overestimate it, potentially leading to overly risky or overly conservative decisions. To address this issue, we develop a parametric bootstrap procedure that is strongly asymptotically consistent and provides a controlled overestimation of entropic risk under mild assumptions. The method first fits a distribution to the data and then estimates the empirical estimator's bias via bootstrapping. We show that the fitted distribution must satisfy only weak regularity conditions, and Gaussian mixture models offer a convenient and flexible choice within this class. As an application, we introduce a distributionally robust optimization model for an insurance contract design problem that incorporates correlations in household losses. We show that selecting regularization parameters using standard cross-validation can lead to substantially higher out-of-sample risk for the insurer if the validation bias is not corrected. Our approach improves performance by recommending higher and more accurate premiums, thereby better reflecting the underlying tail risk.

翻译：熵风险度量因其能捕捉不确定损失相关的尾部风险，在经济学、管理科学、金融以及安全关键控制系统中被广泛应用于高风险决策。然而，当数据有限时，通过用样本均值替代风险度量中的期望所构建的经验熵风险估计量会低估真实风险。我们证明，对于具有无界右尾的分布，这种负偏差随损失标准差的增长呈超线性趋势。我们进一步表明，针对经验风险开发的几种现有偏差修正技术要么继续低估熵风险，要么显著高估熵风险，可能导致决策过于冒险或过于保守。为解决此问题，我们提出了一种参数化自助法程序，该程序具有强渐近一致性，并在温和假设下提供对熵风险的可控高估。该方法首先对数据拟合一个分布，然后通过自助法估计经验估计量的偏差。我们证明拟合分布仅需满足较弱的正则性条件，而高斯混合模型为此类分布提供了一个便捷灵活的选择。作为应用，我们针对包含家庭损失相关性的保险合同设计问题，引入了一个分布鲁棒优化模型。研究表明，若未校正验证偏差，使用标准交叉验证选择正则化参数可能导致保险商的样本外风险显著升高。我们的方法通过推荐更高且更准确的保费来提升性能，从而更好地反映潜在的尾部风险。