  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 data sets of daily precipitation.
