Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mitigating optimistic bias in entropic risk estimation and optimization

Utsav Sadana, Erick Delage, Angelos Georghiou

TL;DR

This work tackles the underestimation of the entropic risk measure in finite samples by showing that bias grows with loss variance, especially for heavy-tailed distributions. It introduces a bias-aware parametric bootstrap framework that fits a flexible Gaussian mixture model to data and uses bootstrap bias estimates to construct conservative, strongly consistent corrections, with overestimation controlled under mild tail assumptions. Three fitting strategies—risk matching, extremal-tail matching, and EVT-inspired tail fitting—are developed to ensure robustness and tractability, enabling closed-form entropic risk calculations within the bootstrap loop. The methods are integrated into entropic risk optimization and a distributionally robust insurance pricing model that handles correlated household losses, and numerical experiments demonstrate improved out-of-sample risk and more accurate premium design compared with traditional cross-validation. Overall, the work provides practical, theory-backed tools for reliable tail-risk estimation and decision-making in finance, insurance, and risk-sensitive control, especially under data scarcity and correlation structures.

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.

Mitigating optimistic bias in entropic risk estimation and optimization

TL;DR

This work tackles the underestimation of the entropic risk measure in finite samples by showing that bias grows with loss variance, especially for heavy-tailed distributions. It introduces a bias-aware parametric bootstrap framework that fits a flexible Gaussian mixture model to data and uses bootstrap bias estimates to construct conservative, strongly consistent corrections, with overestimation controlled under mild tail assumptions. Three fitting strategies—risk matching, extremal-tail matching, and EVT-inspired tail fitting—are developed to ensure robustness and tractability, enabling closed-form entropic risk calculations within the bootstrap loop. The methods are integrated into entropic risk optimization and a distributionally robust insurance pricing model that handles correlated household losses, and numerical experiments demonstrate improved out-of-sample risk and more accurate premium design compared with traditional cross-validation. Overall, the work provides practical, theory-backed tools for reliable tail-risk estimation and decision-making in finance, insurance, and risk-sensitive control, especially under data scarcity and correlation structures.

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.
Paper Structure (62 sections, 38 theorems, 199 equations, 14 figures, 8 algorithms)

This paper contains 62 sections, 38 theorems, 199 equations, 14 figures, 8 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Notations:
  4. Properties of entropic risk measure
  5. Asymptotic effect of variance on estimation bias
  6. Asymptotic effect of variance on empirical risk estimator
  7. Asymptotic effect of variance on common bias correction procedures
  8. Bias correction based on central limit theorem (CLT)
  9. Non-parametric bootstrapping
  10. Leave-one out cross validation
  11. Bias mitigation using bias-aware parametric bootstrapping
  12. The bias mitigation procedure
  13. Q N is a Gaussian Mixture Model
  14. Learning GMM by Maximum Likelihood Estimation
  15. Learning GMM by Entropic Risk Matching
  16. ...and 47 more sections

Key Result

Lemma 1

Under Assumption assum:tail_bound2, ${\mathbb E}[\exp(\alpha \ell(\boldsymbol{z}, \boldsymbol{{\eta}}))]\in \left[\exp(-\frac{G}{C}),\,\frac{G}{C-1}+1\right]$ and $\text{Var}[\exp(\alpha \ell(\boldsymbol{z}, \boldsymbol{{\eta}}))]\in \left[0,\,\frac{2G}{C-2}+1\right]$.

Figures (14)

  • Figure 1: Statistics of the empirical risk for different values of $z \in \{0.6, 0.7, 0.8, 0.9, 1\}$ and training sample sizes $N\in \{50, 100, 200, 500\}$ over $10000$ repetitions. The true risk is given by $(-15/2)\log(1-0.48z)$.
  • Figure 2: Statistics of bias correction estimated from non-parametric bootstrap and parametric bootstrap obtained by fitting a GMM by MLE (BS-MLE), entropic risk matching (BS-Match) and tail fitting (BS-EVT) followed by bootstrapping over $100$ resampling from the underlying distribution.
  • Figure 3: Statistics of the estimates of the true entropic risk obtained from different models for each project.
  • Figure 4: Comparison of the effects of training sample size $N$ on out-of-sample entropic risk (left) and optimal radius $\epsilon^*$ (right). Boxplots present the statistics after $100$ resampling of datasets, and diamonds present the mean for each $N$.
  • Figure 5: Statistics of entropic risk estimators for different radius and $N=1000$ after $100$ resampling of datasets
  • ...and 9 more figures

Theorems & Definitions (45)

  • Example 1
  • Definition 1
  • Lemma 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Definition 2
  • Proposition 5
  • Remark 1
  • Proposition 6
  • ...and 35 more