Asmussen and Lehtomaa [Distinguishing log-concavity from heavy tails. Risks 5(10), 2017] introduced an interesting function $g$ which is able to distinguish between log-convex and log-concave tail behaviour of distributions, and proposed a randomized estimator for $g$. In this paper, we show that $g$ can also be seen as a tool to detect gamma distributions or distributions with gamma tail. We construct a more efficient estimator $\hat{g}_n$ based on $U$-statistics, propose several estimators of the (asymptotic) variance of $\hat{g}_n$, and study their performance by simulations. Finally, the methods are applied to several real data sets.

翻译：Asmussen 与 Lehtomaa [《区分对数凹性与重尾分布》。Risks 5(10), 2017] 引入了一个有趣的函数 $g$，该函数能够区分分布的尾部行为是对数凸还是对数凹，并提出了一种针对 $g$ 的随机化估计量。本文证明 $g$ 同样可作为检测伽马分布或具有伽马尾部特征的分布的工具。我们基于 $U$-统计量构造了一个更高效的估计量 $\hat{g}_n$，提出了 $\hat{g}_n$ 的（渐近）方差的多项估计量，并通过模拟实验评估其性能。最后，将该方法应用于多个实际数据集。