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 ${\xi }_{\star }^2={\sup }_{\lambda \in \mathbb{R}}L(\lambda )$ where $L(\lambda )=\frac{2}{{\lambda }^2}\log \mathbb{E}{e}^{\lambda X}$ is a weighted cumulant generating function. We study the estimation of ${\xi }_{\star }^2$ and prove that the minimax risk is governed by a non-increasing function ${\delta }_P(C)={\sup }_{|\lambda |\ge C}L(\lambda )-{\sup }_{|\lambda |\le C}L(\lambda )$ which captures the influence of the tail behavior of the distribution $P$. Over the class of distributions with ${\delta }_P\le r$ 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 $r((\log n{)}^{1/2-\epsilon })+n^{-1/2+\epsilon }$ for any $\epsilon >0$. 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.

arXiv comment30 pages, 3 figures, and 1 table