Sub-Gaussian Confidence Intervals for Heavy-Tailed Data: Characterizing the Limits of Inference
Abstract
We investigate the problem of uncertainty quantification in the mean estimation framework. Given an i.i.d. sample from a distribution with unknown mean μ and variance σ2, we want to construct an estimator whμn and provide a computable and size-optimal upper bound for the error |wh μn − μ|. While estimators possessing strong deviation guarantees are well known, no data-dependent non-asymptotic upper bounds for their performance exist in general. We show that this gap can be bridged by introducing a parameter that characterizes the ``effective sample size'' available for uncertainty quantification. This parameter captures the difficulty of the problem, interpolating between ``easy'' cases where sub-Gaussian confidence intervals exist and ``hard" cases where their construction is impossible. Using this characterization, we design confidence intervals of optimal length that are fully adaptive to the unknown variance. Numerical experiments confirm that our approach maintains nominal coverage even in asymmetric and heavy-tailed regimes where other existing methods fail.
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BibTeX
@misc{hammouda2026subgaussian,
title = {{Sub-Gaussian Confidence Intervals for Heavy-Tailed Data: Characterizing the Limits of Inference}},
author = {Ilyes Hammouda and Stanislav Minsker and Mohamed Ndaoud},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15799},
eprint = {2610.15799},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15799},
}
RIS
TY - UNPB
TI - Sub-Gaussian Confidence Intervals for Heavy-Tailed Data: Characterizing the Limits of Inference
AU - Ilyes Hammouda
AU - Stanislav Minsker
AU - Mohamed Ndaoud
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15799
UR - https://arxiv.org/abs/2610.15799
AB - We investigate the problem of uncertainty quantification in the mean estimation framework. Given an i.i.d. sample from a distribution with unknown mean μ and variance σ^2, we want to construct an estimator whμ_n and provide a computable and size-optimal upper bound for the error |wh μ_n − μ|. While estimators possessing strong deviation guarantees are well known, no data-dependent non-asymptotic upper bounds for their performance exist in general. We show that this gap can be bridged by introducing a parameter that characterizes the ``effective sample size'' available for uncertainty quantification. This parameter captures the difficulty of the problem, interpolating between ``easy'' cases where sub-Gaussian confidence intervals exist and ``hard" cases where their construction is impossible. Using this characterization, we design confidence intervals of optimal length that are fully adaptive to the unknown variance. Numerical experiments confirm that our approach maintains nominal coverage even in asymmetric and heavy-tailed regimes where other existing methods fail.
ER -