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On sample ranges from two sets of heterogenous random variables

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  • Ding, Weiyong
  • Da, Gaofeng
  • Zhao, Peng

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].

Suggested Citation

  • Ding, Weiyong & Da, Gaofeng & Zhao, Peng, 2013. "On sample ranges from two sets of heterogenous random variables," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 63-73.
    • Handle: RePEc:eee:jmvana:v:116:y:2013:i:c:p:63-73
    DOI: 10.1016/j.jmva.2012.11.009
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    References listed on IDEAS

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    1. Proschan, F. & Sethuraman, J., 1976. "Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability," Journal of Multivariate Analysis, Elsevier, vol. 6(4), pages 608-616, December.
    2. Genest, Christian & Kochar, Subhash C. & Xu, Maochao, 2009. "On the range of heterogeneous samples," Journal of Multivariate Analysis, Elsevier, vol. 100(8), pages 1587-1592, September.
    3. Zhao, Peng & Li, Xiaohu & Balakrishnan, N., 2009. "Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 952-962, May.
    4. Zhao, Peng & Zhang, Yiying, 2012. "On sample ranges in multiple-outlier models," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 335-349.
    5. Xu, Maochao & Balakrishnan, N., 2012. "On the sample ranges from heterogeneous exponential variables," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 1-9.
    6. Kochar, Subhash C & Korwar, Ramesh, 1996. "Stochastic Orders for Spacings of Heterogeneous Exponential Random Variables," Journal of Multivariate Analysis, Elsevier, vol. 57(1), pages 69-83, April.
    7. Kochar, Subhash & Rojo, Javier, 1996. "Some New Results on Stochastic Comparisons of Spacings from Heterogeneous Exponential Distributions," Journal of Multivariate Analysis, Elsevier, vol. 59(2), pages 272-281, November.
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