Mitigating optimistic bias in entropic risk estimation and optimization

Abstract

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.