A gamma tail statistic and its asymptotics

06/13/2023

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by   Toshiya Iwashita, et al.

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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 ĝ_n based on U-statistics, propose several estimators of the (asymptotic) variance of ĝ_n, and study their performance by simulations. Finally, the methods are applied to several real data sets.