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On maximum order statistics from heterogeneous geometric variables

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  • Peng Zhao
  • Feng Su

Abstract

Let X 1 ,X 2 be independent geometric random variables with parameters p 1 ,p 2 , respectively, and Y 1 ,Y 2 be i.i.d. geometric random variables with common parameter p. It is shown that X 2:2 , the maximum order statistic from X 1 ,X 2 , is larger than Y 2:2 , the second order statistic from Y 1 ,Y 2 , in terms of the hazard rate order [usual stochastic order] if and only if $p\geq \tilde{p}$ , where $\tilde{p}=(p_{1}p_{2})^{\frac{1}{2}}$ is the geometric mean of (p 1 ,p 2 ). This result answers an open problem proposed recently by Mao and Hu (Probab. Eng. Inf. Sci. 24:245–262, 2010) for the case when n=2. Copyright Springer Science+Business Media, LLC 2014

Suggested Citation

  • Peng Zhao & Feng Su, 2014. "On maximum order statistics from heterogeneous geometric variables," Annals of Operations Research, Springer, vol. 212(1), pages 215-223, January.
    • Handle: RePEc:spr:annopr:v:212:y:2014:i:1:p:215-223:10.1007/s10479-012-1158-6
    DOI: 10.1007/s10479-012-1158-6
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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. Jeske, Daniel R. & Blessinger, Todd, 2004. "Tunable Approximations for the Mean and Variance of the Maximum of Heterogeneous Geometrically Distributed Random Variables," The American Statistician, American Statistical Association, vol. 58, pages 322-327, November.
    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.
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