TITLE:
Simultaneous Confidence Bands for Conditional Risk Measurement and Conditional Expected Loss Based on Generalized Estimators
AUTHORS:
Jiale Diao
KEYWORDS:
Conditional Risk Measures, Simultaneous Confidence Bands, Extreme Value Theory, Multiplier Bootstrap, Heavy Tails
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.6,
June
22,
2026
ABSTRACT: Accurate measurement and valid inference of extreme financial risks remain a fundamental challenge in risk management research. Existing literature on conditional risk measures predominantly focuses on pointwise estimation and confidence intervals for single tail thresholds, failing to support joint statistical inference over the entire extreme tail interval. Meanwhile, traditional asymptotic-based confidence bands suffer from severe finite-sample coverage bias, and the complex limiting Gaussian process under the intermediate asymptotic scenario impedes direct empirical application. Under the location-scale model framework, this paper proposes a multiplier bootstrap approach to construct simultaneous confidence bands (SCBs) for conditional risk measures of Pareto-type heavy-tailed return series. We adopt the generalized moment Hill estimator and select effective sample sizes via a data-driven criterion, with rigorous proof of the bootstrap procedure’s asymptotic validity. Monte Carlo simulations confirm that the proposed method is highly robust to the tail index estimator’s moment order
α
, with coverage probabilities converging to the 95% nominal level and confidence band lengths shrinking at the rate governed by the effective tail sample size, consistent with the theoretical
k
?1/2
rate. The method delivers superior coverage accuracy for core relative risk measures, effectively mitigating finite-sample bias in traditional approaches, balancing theoretical rigor and practical tractability for financial risk management applications.