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arXiv · 2511.05840

Comparative e-backtests for general risk measures

Zhanyi Jiao·Qiuqi Wang·Yimiao Zhao
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Abstract

Backtesting risk measures is a central task in financial regulation. While standard backtests evaluate whether a forecasting model is statistically consistent with observed losses, regulatory practice often requires assessing the performance of an internal model relative to benchmark models. We develop a non-parametric sequential framework for comparative backtests of general elicitable risk measures using e-values and e-processes. The proposed methods provide anytime-valid inference and remain robust under dependence and model misspecification. In particular, we propose a modified three-zone approach based on weak dominance, which yields more informative conclusions in comparative backtesting. As a technical building block, we also construct general standard e-backtests for identifiable risk measures and characterize the associated e-values and e-processes. The resulting procedures apply to a broad class of commonly used risk measures, including the mean, variance, Value-at-Risk, Expected Shortfall, and expectiles. Simulation studies and empirical analyses illustrate the effectiveness of the proposed approach.

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Zhanyi Jiao, Qiuqi Wang, Yimiao Zhao. 2025-11-08. Comparative e-backtests for general risk measures. https://arxiv.org/abs/2511.05840

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