Year: 2026
Type: preprint
Abstract: The sub-Gaussian parameter (also called the variance proxy) of a mean-zero random variable $X$ is defined as $ξ^2_\star = \sup_{λ\in \mathbb{R}} L(λ)$ where $L(λ) = \frac{2}{λ^2} \log \mathbb{E} e^{λX}$ is a weighted cumulant generating function. We study the estimation of $ξ^2_\star$ and prove that the minimax risk is governed by a non-increasing function $δ_P(C) = \sup_{|λ| \geq C} L(λ) - \sup_{|λ| \leq C} L(λ)$ which captures the influence of the tail behavior of the distribution $P$. Over the class of distributions with $δ_P \leq r$ for a non-increasing function $r$, the minimax risk is, up to a multiplicative constant, lower bounded by $r(\sqrt{\log n}) + n^{-1/2}$ and upper bounded by $r((\log n)^{1/2-\varepsilon}) + n^{-1/2 + \varepsilon}$ for any $\varepsilon > 0$. 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.
Source: arXiv (Cornell University)
Authors: Liu, Jason, Xu, Min, Xing, Jinchuan
Institution: Rutgers, The State University of New Jersey
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7163687366