On sample ranges from two sets of heterogenous random variables

Authors:
Weiyong Ding;Gaofeng Da;Peng Zhao
Affiliations:
College of Science, Hebei United University, Tangshan 063009, China;Department of Statistics and Finance, University of Science and Technology of China, Hefei 230026, China;School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China
Venue:
Journal of Multivariate Analysis
Year:
2013

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Abstract

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].