A gamma tail statistic and its asymptotics (Q6555329)

Given independent random variables \(X_1\) and \(X_2\) with a common distribution, the authors consider the function \(g:(0,\infty)\to[0,1]\) defined by \textit{S. Asmussen} and \textit{J. Lehtomaa} [Risks 5, No. 1, Paper No. 10, 14 p. (2017; \url{doi:10.3390/risks5010010})] as follows:\N\[\Ng(d)=\mathbb{E}\left[\left.\frac{|X_1-X_2|}{X_1+X_2}\right|X_1+X_2>d\right]\,.\N\]\NThe authors begin by showing that \(g\) is constant if and only if the underlying distribution is Gamma, in which case the constant value of \(g\) depends on the shape parameter of this Gamma distribution. With the aim of using \(g\) to test for a Gamma distribution or Gamma tail behaviour in IID data, the authors define a new estimator for \(g\) based on \(U\)-statistics, and show that this estimator has an asymptotic Normal distribution. Several methods of estimating the variance in this asymptotic distribution are discussed, and numerical studies are used to illustrate their performance. The paper concludes with an application to precipitation data.