On sample ranges from two sets of heterogenous random variables

As one of the criteria for comparing variabilities among distributions, the sample range has attracted considerable attention in past decades. In this paper, we establish stochastic comparison results of sample ranges arising from two sets of heterogeneous exponential samples. It is shown that the reversed hazard rate of the sample range is a Schur-convex function of the parameter vector while its distribution function is a Schur-concave function of the vector of logarithms of the coordinates of the parameter vector. Moreover, when samples follow the proportional hazard rates models, we prove that the distribution function of the sample range is Schur-concave in the parameter vector, thereby extending several results known in the literature including Kochar and Rojo (1996) [13], Kochar and Xu (2007) [14] and Zhao and Zhang (2012)  [31].