A gamma tail statistic and its asymptotics

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 behavior 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 ̂ gn 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 datasets of daily precipitation.