Large and Moderate Deviations for Conservative Tail-Index Estimation
Abstract
To design systems that are protected against events much rarer than the observational record, extreme-value methods are needed to extrapolate distribution tails. Tail-index estimators such as the Hill estimator are central to this extrapolation, but overestimating the tail exponent can lead to substantial underestimation of rare-event probabilities. Motivated by this, we derive large- and moderate-deviation asymptotics for the Hill estimator and use them to construct estimators whose probability of exceeding the true tail index decays at a controlled exponential rate (the decay rate). In the large-deviations regime, we show that a simple rescaled version of the Hill estimator achieves an optimal balance between bias and decay rate among scale-invariant estimators based on the same top k order statistics. Under a second-order condition, we quantify the effect of the Hill bias, analyze a bias-corrected estimator, and identify sufficient conditions for moderate deviations in the boundary case where the second-order parameter ρ equals zero.
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BibTeX
@misc{gosgens2026large,
title = {{Large and Moderate Deviations for Conservative Tail-Index Estimation}},
author = {Martijn Gösgens and Bart P. G. van Parys and Bert Zwart},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31127},
eprint = {2609.31127},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31127},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Large and Moderate Deviations for Conservative Tail-Index Estimation
AU - Martijn Gösgens
AU - Bart P. G. van Parys
AU - Bert Zwart
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31127
UR - https://arxiv.org/abs/2609.31127
AB - To design systems that are protected against events much rarer than the observational record, extreme-value methods are needed to extrapolate distribution tails. Tail-index estimators such as the Hill estimator are central to this extrapolation, but overestimating the tail exponent can lead to substantial underestimation of rare-event probabilities. Motivated by this, we derive large- and moderate-deviation asymptotics for the Hill estimator and use them to construct estimators whose probability of exceeding the true tail index decays at a controlled exponential rate (the decay rate). In the large-deviations regime, we show that a simple rescaled version of the Hill estimator achieves an optimal balance between bias and decay rate among scale-invariant estimators based on the same top k order statistics. Under a second-order condition, we quantify the effect of the Hill bias, analyze a bias-corrected estimator, and identify sufficient conditions for moderate deviations in the boundary case where the second-order parameter ρ equals zero.
ER -