Valid data-driven uncertainty quantification for heavy-tailed rare-event probabilities of sums under input tail uncertainty

Estimating an exceedance probability such as $p = P(S_n > \gamma)$, the chance that a sum of $n$ independent input quantities crosses a high threshold, is a basic task of risk analysis, but when the summand distribution must be learned from data the estimate inherits input uncertainty concentrated in the distribution's tail. Huang and Lam (2019) proved a dichotomy — truncating the input tail destroys heavy-tailed rare-event estimates (relative error approaching 100%) while barely affecting light-tailed ones — and showed experimentally that bootstrap confidence intervals for $p$ have near-zero coverage under heavy tails, with generalized Pareto tail fitting improving coverage but leaving a misspecification bias that dominates as the data size grows. The authors' full journal treatment (Huang, Lam, and Liu, 2023, arXiv preprint) supplied the deferred proofs, data-size thresholds, and a tail-heaviness detection heuristic, yet the statistical core remains open: no available method gives provably valid confidence intervals for heavy-tailed rare-event probabilities of sums from finite data without essentially correct parametric tail assumptions, and both the theory and the methodology remain confined to independent, identically distributed summands and to exceedance probabilities. The open problem is to determine whether such an interval, robust to the slowly varying tail component that misspecifies any fitted generalized Pareto model, can exist — and if so to construct it, otherwise to prove a matching impossibility; characterizing the data size needed for validity and extending the analysis to dependent inputs or to risk measures such as Value-at-Risk and expected shortfall are further open questions.

Background

Assessing the probability of rare, catastrophic outcomes — large portfolio losses, ruin of an insurance portfolio, system overload — often reduces to estimating an exceedance probability $p = P(S_n > \gamma)$, where $S_n = X_1 + \cdots + X_n$ aggregates $n$ input random variables and $\gamma$ is a high threshold. In simulation-based analysis the summand distribution $F$ is itself calibrated from data, and the error propagated from that calibration is called input uncertainty, as distinct from Monte Carlo noise, the sampling error of the simulation itself, which can be made negligible by variance reduction. The most data-driven choice replaces $F$ by the empirical distribution of $N$ observations; the nonparametric bootstrap then resamples the data to build a confidence interval (CI) for $p$, and the coverage of a CI is the probability, over the randomness of the data, that it contains the true value.

Everything hinges on how much the data say about the tail of $F$. A distribution is light-tailed, here, if it possesses finite exponential moments, meaning the logarithmic moment generating function $\psi(\theta) = \log E[e^{\theta X}]$ is finite near $\theta = 0$; then $p$ decays exponentially in $n$ by Cramér-type large deviations, because the rare event is reached by every summand shifting a little. A distribution is heavy-tailed, here, if its tail is regularly varying, $\bar F(x) = L(x)x^{-\alpha}$ with tail index $\alpha$ and a slowly varying function $L$ (one for which $L(tx)/L(x) \to 1$ for every $t > 1$), or more broadly subexponential — the class for which the tail of a sum is asymptotically the tail of its maximum, so the rare event happens through one "big jump" of a single summand. Huang and Lam (2019) proved a dichotomy for the effect of truncating the input tail at a level $u$ (a stand-in for the reach of the data): for heavy tails with $u \le (\gamma - n\mu)/\sqrt{\log n}$, the truncated-input estimate misses $p$ by essentially its full magnitude, $p(\tilde F_u) - p(F) = -p(F)(1 + o(1))$, while for light tails the error is negligible as soon as $n e^{\theta^* u}\bar F(u) \to 0$, a condition met at only logarithmic levels $u$.

Consequently the plug-in estimate and the bootstrap, which carry no mass beyond the data, can be meaningless for heavy-tailed inputs. In the source experiments with Gaussian summands the bootstrap CIs attain near-nominal coverage, but with heavy-tailed summands their measured coverage is close to zero even with $N = 10^7$ observations for a probability of order $10^{-7}$, and most intervals have zero width. Fitting a generalized Pareto distribution (GPD) to the data above a high threshold — the peak-over-threshold method, justified because threshold exceedances of a wide class of distributions are approximately GPD — and bootstrapping with the fitted tail improves coverage, but the remaining bias, stemming from the unknown slowly varying component $L$ that no fitted GPD captures, dominates as $N$ grows and coverage can again collapse to zero.

The authors' full journal treatment (Huang, Lam, and Liu 2023) supplied the deferred proofs of this dichotomy, including the variant for the non-strict inequality $S_n \ge \gamma$ and experiments on non-power-law subexponential tails such as lognormal, and it derived data-size thresholds: reliable estimation of a heavy-tailed $p$ requires the data size to grow polynomially in the number of summands $n$ (roughly, larger than $n/p$), versus only logarithmically for light tails. It also proposed detecting the under-estimation risk with extreme value index estimators, which estimate the GPD shape parameter whose sign classifies light versus heavy tails. Parallel methodological lines address tail model risk by other means: worst-case tail bounds without parametric models (Lam and Mottet 2017), distributionally robust inference for extreme Value-at-Risk (Yuen, Stoev, and Cooley 2020), and worst-case rare-event probabilities over Wasserstein ambiguity sets (Ahn, Chen, and Zheng 2026). The input-uncertainty literature has also improved bootstrap resampling for general output measures (Song, Lam, and Barton 2024; Chen, Lam, and Liu 2024), and separate work constructs confidence intervals for rare-event estimators subject to Monte Carlo noise alone (Bai and Lam 2020; Bai and Lam 2025). None of these provides a data-driven interval with valid coverage for the probability of a heavy-tailed sum, and the dichotomy analysis itself remains confined to independent, identically distributed (i.i.d.) summands and to exceedance probabilities.

Problem Statement

The problem, as posed by the source's open issues and left unresolved by its journal version, is to make data-driven rare-event estimation statistically trustworthy when the input tail is heavy and unknown. For the exceedance probability $p = P(S_n > \gamma)$, where $S_n = X_1 + \cdots + X_n$ sums independent, identically distributed summands of unknown distribution $F$ observed only through $N$ data points, and for the heavy-tailed regime where $F$ is subexponential — including regularly varying tails $\bar F(x) = L(x)x^{-\alpha}$ with arbitrary unknown slowly varying $L$ — the open questions are as follows:

1. (Uncertainty quantification) Construct a data-driven estimator of $p$ together with a confidence interval whose coverage at a stated nominal level is provably valid, finite-sample or asymptotically, uniformly over the stated class of summand distributions, without assuming a correctly specified parametric tail family. This must overcome the two demonstrated failure modes: the collapse of the plain bootstrap under heavy tails, and the growing dominance, as $N$ increases, of the misspecification bias that the unknown slowly varying component induces in any fitted generalized Pareto tail. Whether such a data-driven interval can exist without stronger tail assumptions — and which stronger tail assumptions would be required if it cannot — is part of the same question.

2. (Data requirement) Characterize how the data size needed for validity scales with the number of summands $n$, the rarity level of the event, and the tail heaviness, refining the coarse existing guidance that the data size must be significantly larger than $n/p$.

3. (Generalizations) Extend the tail-uncertainty dichotomy and the valid uncertainty-quantification methodology beyond the basic i.i.d. exceedance-probability setting: to dependent or non-identically distributed summands, and to other rare-event risk measures such as Value-at-Risk (a tail quantile) and expected shortfall (the conditional expectation of the loss given that it exceeds that quantile).

These components jointly serve one overarching objective: knowing when a data-driven rare-event probability estimate and its error bar can be believed, and when they instead catastrophically understate risk.

References

1. Zhiyuan Huang and Henry Lam (2019), "On The Impacts of Tail Model Uncertainty in Rare-Event Estimation", Proceedings of the 2019 Winter Simulation Conference, pp. 950–961. DOI: 10.1109/WSC40007.2019.9004933, https://doi.org/10.1109/WSC40007.2019.9004933 (open-access full text: https://www.informs-sim.org/wsc19papers/102.pdf)
2. Zhiyuan Huang, Henry Lam, and Zhenyuan Liu (2023), "Propagation of Input Tail Uncertainty in Rare-Event Estimation: A Light versus Heavy Tail Dichotomy", arXiv:2401.00172. DOI: 10.48550/arXiv.2401.00172, https://arxiv.org/abs/2401.00172
3. Henry Lam and Clementine Mottet (2017), "Tail Analysis Without Parametric Models: A Worst-Case Perspective", Operations Research 65(6):1696–1711. DOI: 10.1287/opre.2017.1643, https://doi.org/10.1287/opre.2017.1643
4. Robert Yuen, Stilian Stoev, and Daniel Cooley (2020), "Distributionally robust inference for extreme Value-at-Risk", Insurance: Mathematics and Economics 92:70–89. DOI: 10.1016/j.insmatheco.2020.03.003, https://doi.org/10.1016/j.insmatheco.2020.03.003
5. Gebhard Geiger (2024), "Catastrophic risk: indication, quantitative assessment and management of rare extreme events using a non-expected utility framework", Annals of Operations Research 343:223–261. DOI: 10.1007/s10479-024-06259-z, https://doi.org/10.1007/s10479-024-06259-z
6. Dohyun Ahn, Huiyi Chen, and Lewen Zheng (2026), "Wasserstein Distributionally Robust Rare-Event Simulation", arXiv:2601.01642. DOI: 10.48550/arXiv.2601.01642, https://arxiv.org/abs/2601.01642
7. Yuanlu Bai and Henry Lam (2020), "On the Error of Naive Rare-Event Monte Carlo Estimator", Proceedings of the 2020 Winter Simulation Conference, pp. 397–408. DOI: 10.1109/WSC48552.2020.9383905, https://doi.org/10.1109/WSC48552.2020.9383905
8. Yuanlu Bai and Henry Lam (2025), "Uncertainty Quantification and Confidence Intervals for Naive Rare-Event Estimators", Journal of Applied Probability 62(1):84–110. DOI: 10.1017/jpr.2024.43, https://doi.org/10.1017/jpr.2024.43 (preprint: arXiv:2305.02434, https://arxiv.org/abs/2305.02434)
9. Motong Chen, Henry Lam, and Zhenyuan Liu (2024), "Quantifying Distributional Input Uncertainty via Inflated Kolmogorov-Smirnov Confidence Band", arXiv:2403.09877. DOI: 10.48550/arXiv.2403.09877, https://arxiv.org/abs/2403.09877
10. Eunhye Song, Henry Lam, and Russell R. Barton (2024), "A Shrinkage Approach to Improve Direct Bootstrap Resampling Under Input Uncertainty", INFORMS Journal on Computing 36(4):1023–1039. DOI: 10.1287/ijoc.2022.0044, https://doi.org/10.1287/ijoc.2022.0044