In non-asymptotic statistical inferences, variance-type parameters of sub-Gaussian distributions play a crucial role. However, direct estimation of these parameters based on the empirical moment generating function (MGF) is infeasible. To this end, we recommend using a sub-Gaussian intrinsic moment norm [Buldygin and Kozachenko (2000), Theorem 1.3] through maximizing a series of normalized moments. Importantly, the recommended norm can not only recover the exponential moment bounds for the corresponding MGFs, but also lead to tighter Hoeffding's sub-Gaussian concentration inequalities. In practice, {\color{black} we propose an intuitive way of checking sub-Gaussian data with a finite sample size by the sub-Gaussian plot}. Intrinsic moment norm can be robustly estimated via a simple plug-in approach. Our theoretical results are applied to non-asymptotic analysis, including the multi-armed bandit.

翻译：在非渐近统计推断中，次高斯分布的方差类参数起着关键作用。然而，基于经验矩生成函数直接估计这些参数是不可行的。为此，我们推荐使用通过最大化一系列归一化矩得到的次高斯内在矩范数[Buldygin 与 Kozachenko (2000)，定理 1.3]。重要的是，该推荐的范数不仅能够恢复对应矩生成函数的指数矩界，还能导出更紧致的霍夫丁次高斯浓度不等式。在实际操作中，我们提出了一种直观方法，通过次高斯图来检验有限样本量下的次高斯数据。内在矩范数可通过简单的插件方法进行稳健估计。我们的理论结果被应用于非渐近分析，包括多臂老虎机问题。