{"ID":3083596,"CreatedAt":"2026-06-05T06:46:15.197025399Z","UpdatedAt":"2026-06-07T05:16:48.22291569Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.06384","arxiv_id":"2606.06384","title":"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 $ξ^2_* = \\sup_{λ\\in \\mathbb{R}} L(λ)$ where $L(λ) = \\frac{2}{λ^2} \\log \\mathbb{E} e^{λX}$ is a weighted cumulant generating function. Despite the ubiquity of sub-Gaussian random variables, the estimation of $ξ^2_*$ has received little attention and is not yet well understood. In this work, we study a natural estimator of $ξ^2_*$ based on constrained maximization of the empirical analogue of $L$. We prove that the estimator is consistent bound the rates of convergence under assumptions on $L$: if $L$ has an maximizer, then our bound is $O_p(n^{-1/2 + \\varepsilon})$ for any $\\varepsilon \u003e 0$; if the argmax of $L$ is also bounded, then the bound improves to $O_p(n^{-1/2})$. We show that our assumptions on $L$ are necessary by proving that the minimax risk over all sub-Gaussian distributions is $Ω(1)$; imposing increasingly strong assumptions on the tail growth of $L$ yields a continuum of classes whose minimax lower bound interpolates between $Ω(1/\\log n)$ and $Ω(1)$. Root-n rate is possible if we restrict to a subclass of distributions where $L$ attains its supremum in a bounded region, in which case our estimator is minimax optimal. If the underlying distribution is not sub-Gaussian, we show that our estimator goes to infinity with a divergence rate controlled by the tail of the distribution. Finally, we apply our estimator in a Gene Ontology (GO) enrichment study to construct p-values for a large-scale permutation test, showing that it can serve as a reliable alternative to the peaks-over-threshold approach, particularly in regimes where the peaks-over-threshold method is of uncertain validity.","short_abstract":"The sub-Gaussian parameter (also called the variance proxy) of a mean-zero random variable $X$ is defined as $ξ^2_* = \\sup_{λ\\in \\mathbb{R}} L(λ)$ where $L(λ) = \\frac{2}{λ^2} \\log \\mathbb{E} e^{λX}$ is a weighted cumulant generating function. Despite the ubiquity of sub-Gaussian random variables, the estimation of $ξ^2...","url_abs":"https://arxiv.org/abs/2606.06384","url_pdf":"https://arxiv.org/pdf/2606.06384v1","authors":"[\"Jason Liu\",\"Min Xu\",\"Jinchuan Xing\"]","published":"2026-06-04T16:48:31Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"stat.ME\",\"stat.ML\"]","methods":"[]","has_code":false}