Valid uncertainty quantification and completion of the truncation-reliability theory for data-driven rare-event estimation under unknown input tails

The estimation of rare-event probabilities $p=P(S_n>\gamma)$ for sums of i.i.d. inputs from finitely many input samples is unreliable in a fundamentally asymmetric way: Huang, Lam and Liu (2023) proved that when the input distribution is heavy-tailed, truncating its tail at any data-informable level drives the relative error of the exceedance probability to $-1$, while light-tailed problems tolerate merely logarithmic truncation. The statistical side of this dichotomy, however, is unfinished: the nonparametric bootstrap badly under-covers in heavy-tailed scenarios, generalized Pareto tail-fitting restores coverage only under approximately correct tail specification, extreme value index estimators detect but cannot rank tail severity, all reliability guarantees are asymptotic, the light-tailed unreliable regime is uncharacterized, and intermediate subexponential tails (log-normal, Weibull with shape parameter $<1$) are supported only by experiments. The problem asks for the completion of this program: a procedure with provably valid, non-degenerate confidence intervals for heavy-tailed rare-event probabilities robust to tail misspecification, together with the supporting theory — non-asymptotic truncation-error bounds, light-tailed unreliability conditions, a rigorous sample-size theory, and truncation analysis for intermediate subexponential tails.

Background

Rare-event estimation concerns very small probabilities such as $p=P(S_n>\gamma)$, where $S_n=X_1+\cdots+X_n$ is a sum of i.i.d. random variables with common distribution $F$ and $\gamma$ is a high threshold — the basic mathematical model behind buffer overflows, insurance ruin, portfolio loss, and system-failure probabilities. Classical rare-event simulation assumes $F$ is known and studies how to compute $p$ efficiently. In data-driven settings, $F$ is unknown and estimated from $N$ i.i.d. samples, for instance by the empirical distribution $\hat F_N$; the error propagated from this input estimation (rather than from Monte Carlo noise) is called input uncertainty or model uncertainty.

Huang, Lam and Liu (2023), extending a Winter Simulation Conference precursor (Huang and Lam 2019), study how the lack of tail information in the data distorts $p$. Their vehicle is tail truncation: replace $F$ by $\tilde F_u$, which conditions on $X\le u$ (so $\tilde F_u$ has no mass beyond the level $u$); using the empirical distribution corresponds to truncating at the sample maximum. Two regimes emerge. If $F$ is heavy-tailed in the sense of regular variation, $\bar F(x)=1-F(x)=x^{-\alpha}L(x)$ with tail index $\alpha>2$, a finite $(2+\delta)$-th moment ($E|X|^{2+\delta}<\infty$ for some $\delta>0$), and $L$ a slowly varying function (one that varies sub-polynomially, like $\log x$), then rare events occur through one or a few 'big jumps', so the far tail of $F$ dominates $p$. Their Theorem 2.1 shows that if $\gamma=n\mu+\omega(\sqrt{n\log n})$ (where $\mu=E[X]$ and $\omega$ denotes a quantity growing faster than its argument) and the truncation level satisfies $u\le\mu+O((\gamma-n\mu)/\sqrt{\log n})$, then $p(\tilde F_u)-p(F)=-p(F)(1+o(1))$: the estimate is asymptotically void (100% relative under-estimation). Theorem 2.2 shows reliability, $p(\tilde F_u)-p(F)=o(p(F))$, once $u=\omega((\gamma-n\mu)^{\beta})$ for some $\beta>1$. If instead $F$ is light-tailed (it has a finite exponential moment, i.e. the log moment generating function $\psi(\theta)=\log E[e^{\theta X}]$ is finite near $0$), rare events occur by many small collective shifts (Cramér-type large deviations with rate function $I=\sup_\theta\{b\theta-\psi(\theta)\}$ for $\gamma=bn$), and Theorem 2.4 shows that a merely logarithmic truncation level suffices for reliability. Converting these thresholds into sample-size guidance via the asymptotic order of the sample maximum $M_N$ (from extreme value theory) gives, for heavy tails, rough criteria such as unreliability for $N\le \bar M^{\alpha}n/(p(F)(\sqrt{\log n})^{\alpha})$ and reliability for $N>n^{\beta}/(p(F))^{\beta}$, with the practical advice to use a sample size confidently larger than $n/p(F)$ (here $\bar M>0$ is the fixed constant appearing in the source's criterion).

On the statistical side, the paper evaluates safeguards. The nonparametric bootstrap (resampling the data and recomputing the estimate) badly under-covers — its confidence intervals fail to contain the truth with the nominal probability — for heavy-tailed inputs even at large $N$. Fitting the tail of each resample by a generalized Pareto distribution (GPD), motivated by the peaks-over-threshold theorem of extreme value theory (excesses over a high threshold are approximately GPD), improves coverage but degrades sharply when the true tail is only approximately Pareto, because the unknown slowly varying factor makes the GPD a misspecified tail model. Extreme value index estimators (Pickands and moment estimators, which estimate the tail index from upper order statistics) reliably detect the presence of heavy-tail risk but cannot rank tail severity accurately. Related worst-case approaches — Lam and Mottet's nonparametric tail bounds (2017), distributionally robust extreme value analysis (Blanchet, He and Murthy 2020; Bai, Lam and Zhang 2023), and the recent Wasserstein distributionally robust rare-event simulation of Ahn, Chen and Zheng (2026) — supply conservative worst-case bounds rather than calibrated intervals. A class intermediate between light and regularly varying tails is the subexponential distributions (those whose sum's tail is asymptotically twice the maximum's tail): log-normal and Weibull with shape parameter $<1$ tails belong, and the source paper treats them only numerically.

Problem Statement

Let $p(F)=P(S_n>\gamma)$ where $S_n=X_1+\cdots+X_n$ is a sum of i.i.d. random variables with unknown distribution $F$, $\gamma=bn$ with $b>E[X]$, and let $\tilde F_u$ denote $F$ truncated at level $u$. The source work established the asymptotic dichotomy: for regularly varying heavy tails with index $\alpha>2$, truncation at any level $u\le\mu+O((\gamma-n\mu)/\sqrt{\log n})$ makes the estimate asymptotically void, while $u=\omega((\gamma-n\mu)^{\beta})$, $\beta>1$, suffices for reliability, and for light tails (finite exponential moment) a logarithmic truncation level suffices. The unresolved problem comprises the following six open issues stated by the source, which this statement unifies under the single objective of a complete data-driven reliability theory for rare-event estimation under unknown input tails. (1) Construct a procedure that attains provably valid confidence intervals for $p(F)$ from $N$ input samples in heavy-tailed scenarios across the board — that is, uniformly over the source's heavy-tailed class of regularly varying tails $\bar F(x)=x^{-\alpha}L(x)$ with index $\alpha>2$ and $E|X|^{2+\delta}<\infty$ for some $\delta>0$ (the intermediate subexponential tails are the separate subject of part (6)), at a prescribed nominal coverage level such as the 95% level used in the source's experiments. Validity here must be non-degenerate: as in the source's own evaluation of candidate intervals, which reports both empirical coverage and interval width relative to the target probability, the interval width must shrink as the sample size $N$ grows, so fixed ranges such as $[0,1]$ — which cover every $p(F)\in(0,1)$ with probability one regardless of the data and hence satisfy coverage vacuously — do not qualify. The current best candidate (bootstrap with generalized Pareto tail fitting) improves coverage but is compromised by tail misspecification, and no method is shown to be valid across all heavy-tailed scenarios. (2) Develop extreme value index estimators that gauge the severity of heavy-tailed behavior — accurately ranking tail heaviness rather than merely detecting its presence, with the ranking criterion stated as a formal property of the estimator (a consistency or error bound on the index estimate, or a guarantee that the produced ordering reflects the true ordering of tail severities, rather than a bare numerical illustration) — so that the source's proposed two-step procedure — first use index estimators to detect heavy-tail risk, then collect input data following the sample-size guidance — can be applied with quantified confidence. (3) Replace the asymptotic theorems by finite-sample guarantees or non-asymptotic bounds on the truncation error $|p(\tilde F_u)-p(F)|/p(F)$ as a function of $n$, $u$, and the tail parameters. (4) Characterize the conditions under which estimation becomes unreliable in the light-tailed case — the analog of the heavy-tailed Theorem 2.1 — including the subtle regimes of bounded or very light tails that the source explicitly leaves open. (5) Develop a rigorous statistical theory connecting sample size $N$, truncation level, and estimation accuracy, replacing the current rough guidelines derived from the asymptotic order of the sample maximum. (6) Extend the truncation-effect theory to intermediate-tailed subexponential distributions, specifically log-normal and Weibull tails with shape parameter $<1$, whose behavior is currently supported only by numerical evidence. The unresolved program consists of all six parts together: each is a requirement in its own right, none is optional, and no single part by itself constitutes the completion the source leaves open. Parts (3) and (5) are independent rather than nested: part (3) concerns a deterministic truncation level $u$ as in the source's asymptotic theorems, whereas part (5) concerns the data-driven truncation level induced by the $N$ input samples, so establishing either does not discharge the other.

References

1. Huang Z., Lam H., Liu Z. (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
2. Huang Z., Lam H. (2019). On The Impacts of Tail Model Uncertainty in Rare-Event Estimation. In Proceedings of the 2019 Winter Simulation Conference (WSC), pp. 950-961, DOI 10.1109/WSC40007.2019.9004933. https://doi.org/10.1109/WSC40007.2019.9004933
3. Bai Y., Lam H. (2025). Uncertainty quantification and confidence intervals for naive rare-event estimators. Journal of Applied Probability 62(1):84-110 (published online September 2024), DOI 10.1017/jpr.2024.43. https://doi.org/10.1017/jpr.2024.43
4. Ahn D., Chen H., Zheng L. (2026). Wasserstein Distributionally Robust Rare-Event Simulation. arXiv:2601.01642. https://arxiv.org/abs/2601.01642
5. Bai Y., Lam H., Zhang X. (2023). A Distributionally Robust Optimization Framework for Extreme Event Estimation. arXiv:2301.01360. https://arxiv.org/abs/2301.01360
6. Chen M., Lam H., Liu Z. (2024). Quantifying Distributional Input Uncertainty via Inflated Kolmogorov-Smirnov Confidence Band. arXiv:2403.09877. https://arxiv.org/abs/2403.09877
7. Geiger G. (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
8. Lam H., Mottet C. (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
9. Blanchet J., He F., Murthy K. (2020). On distributionally robust extreme value analysis. Extremes 23:317-347, DOI 10.1007/s10687-019-00371-1. https://doi.org/10.1007/s10687-019-00371-1
10. Burov S. (2026). A Gaussian-Remainder Hierarchy for Sums of Random Variables with Big-Jump Statistics. arXiv:2607.12357. https://arxiv.org/abs/2607.12357