A gamma tail statistic and its asymptotics

Abstract

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