We propose the notion of sub‐Weibull distributions, which are characterized by tails lighter than (or equally light as) the right tail of a Weibull distribution. This novel class generalizes the sub‐Gaussian and sub‐Exponential families to potentially heavier tailed distributions. Sub‐Weibull distributions are parameterized by a positive tail index θ and reduce to sub‐Gaussian distributions for $\theta =1/2$ and to sub‐Exponential distributions for $\theta =1$. A characterization of the sub‐Weibull property based on moments and on the moment generating function is provided and properties of the class are studied. An estimation procedure for the tail parameter is proposed and is applied to an example stemming from Bayesian deep learning.

中文翻译：

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次魏布尔分布：将次高斯和次指数性质推广到较重的尾部分布

我们提出了次Weibull分布的概念，其特征是尾巴比Weibull分布的右尾巴更轻（或与Weibull分布的右尾巴一样轻）。这个新颖的类将亚高斯族和亚指数族归纳为潜在的重尾分布。次魏布尔分布由正尾指数θ进行参数化，并简化为次高斯分布$\theta =\mathrm{1个}/2$ 以及次指数分布 $\theta =\mathrm{1个}$。提供了基于矩和矩生成函数的Sub-Weibull属性的表征，并研究了该类的属性。提出了针对尾部参数的估计程序，并将其应用于源自贝叶斯深度学习的示例。