Estimation of the sub-Gaussian Parameter
Abstract
The sub-Gaussian parameter (also called the variance proxy) of a mean-zero random variable X is defined as ξ2⋆ = supλ∈ ℝ L(λ) where L(λ) = 2/λ2 log 𝔼 eλX is a weighted cumulant generating function. We study the estimation of ξ2⋆ and prove that the minimax risk is governed by a non-increasing function δP(C) = sup|λ| ≥ C L(λ) − sup|λ| ≤ C L(λ) which captures the influence of the tail behavior of the distribution P. Over the class of distributions with δP ≤ r for a non-increasing function r, the minimax risk is, up to a multiplicative constant, lower bounded by r(√(log n)) + n−1/2 and upper bounded by r((log n)1/2−ε) + n−1/2 + ε for any ε > 0. Our estimator for the upper bound is based on constrained maximization of the empirical analogue of L. In addition to being almost minimax optimal and adaptive, we further prove that the estimator is asymptotic normal under suitable conditions and that, if the underlying distribution is not sub-Gaussian, the estimator diverges with a rate determined by the heaviness of the distributional tail.
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BibTeX
@misc{liu2026estimation,
title = {{Estimation of the sub-Gaussian Parameter}},
author = {Jason Liu and Min Xu and Jinchuan Xing},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.06384},
eprint = {2606.06384},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.06384},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Estimation of the sub-Gaussian Parameter
AU - Jason Liu
AU - Min Xu
AU - Jinchuan Xing
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2606.06384
UR - https://arxiv.org/abs/2606.06384
AB - The sub-Gaussian parameter (also called the variance proxy) of a mean-zero random variable X is defined as ξ^2_⋆ = sup_λ∈ ℝ L(λ) where L(λ) = 2/λ^2 log 𝔼 e^λX is a weighted cumulant generating function. We study the estimation of ξ^2_⋆ and prove that the minimax risk is governed by a non-increasing function δ_P(C) = sup_|λ| ≥ C L(λ) − sup_|λ| ≤ C L(λ) which captures the influence of the tail behavior of the distribution P. Over the class of distributions with δ_P ≤ r for a non-increasing function r, the minimax risk is, up to a multiplicative constant, lower bounded by r(√(log n)) + n^−1/2 and upper bounded by r((log n)^1/2−ε) + n^−1/2 + ε for any ε > 0. Our estimator for the upper bound is based on constrained maximization of the empirical analogue of L. In addition to being almost minimax optimal and adaptive, we further prove that the estimator is asymptotic normal under suitable conditions and that, if the underlying distribution is not sub-Gaussian, the estimator diverges with a rate determined by the heaviness of the distributional tail.
ER -