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Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables

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  • Zhao, Peng
  • Li, Xiaohu
  • Balakrishnan, N.

Abstract

Let X1,...,Xn be independent exponential random variables with respective hazard rates [lambda]1,...,[lambda]n, and let Y1,...,Yn be independent exponential random variables with common hazard rate [lambda]. This paper proves that X2:n, the second order statistic of X1,...,Xn, is larger than Y2:n, the second order statistic of Y1,...,Yn, in terms of the likelihood ratio order if and only if with . Also, it is shown that X2:n is smaller than Y2:n in terms of the likelihood ratio order if and only if These results form nice extensions of those on the hazard rate order in Pa[caron]lta[caron]nea [E. Pa[caron]lta[caron]nea, On the comparison in hazard rate ordering of fail-safe systems, Journal of Statistical Planning and Inference 138 (2008) 1993-1997].

Suggested Citation

  • 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.
  • Handle: RePEc:eee:jmvana:v:100:y:2009:i:5:p:952-962
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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. Lillo, Rosa E. & Nanda, Asok K. & Shaked, Moshe, 2001. "Preservation of some likelihood ratio stochastic orders by order statistics," Statistics & Probability Letters, Elsevier, vol. 51(2), pages 111-119, January.
    3. An, Mark Yuying, 1998. "Logconcavity versus Logconvexity: A Complete Characterization," Journal of Economic Theory, Elsevier, vol. 80(2), pages 350-369, June.
    4. Belzunce, Félix & Ruiz, José M. & Ruiz, M. Carmen, 2002. "On preservation of some shifted and proportional orders by systems," Statistics & Probability Letters, Elsevier, vol. 60(2), pages 141-154, November.
    5. Boland, Philip J. & El-Neweihi, Emad & Proschan, Frank, 1994. "Schur properties of convolutions of exponential and geometric random variables," Journal of Multivariate Analysis, Elsevier, vol. 48(1), pages 157-167, January.
    6. Chang, Kuo-Hwa, 2001. "Stochastic orders of the sums of two exponential random variables," Statistics & Probability Letters, Elsevier, vol. 51(4), pages 389-396, February.
    7. 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.
    8. 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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    Citations

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    Cited by:

    1. Paltanea, Eugen, 2011. "Bounds for mixtures of order statistics from exponentials and applications," Journal of Multivariate Analysis, Elsevier, vol. 102(5), pages 896-907, May.
    2. Wang, Jiantian, 2015. "A stochastic comparison result about hazard rate ordering of two parallel systems comprising of geometric components," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 86-90.
    3. Zhao, Peng & Balakrishnan, N., 2010. "Ordering properties of convolutions of heterogeneous Erlang and Pascal random variables," Statistics & Probability Letters, Elsevier, vol. 80(11-12), pages 969-974, June.
    4. Zhao, Peng & Balakrishnan, N., 2009. "Mean residual life order of convolutions of heterogeneous exponential random variables," Journal of Multivariate Analysis, Elsevier, vol. 100(8), pages 1792-1801, September.
    5. Zhao, Peng & Balakrishnan, N., 2011. "New results on comparisons of parallel systems with heterogeneous gamma components," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 36-44, January.
    6. 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.
    7. Zhao, Peng, 2011. "Some new results on convolutions of heterogeneous gamma random variables," Journal of Multivariate Analysis, Elsevier, vol. 102(5), pages 958-976, May.
    8. Balakrishnan, N. & Zhao, Peng, 2013. "Hazard rate comparison of parallel systems with heterogeneous gamma components," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 153-160.
    9. Zhao, Peng & Balakrishnan, N., 2014. "A stochastic inequality for the largest order statistics from heterogeneous gamma variables," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 145-150.
    10. Ebrahim Amini-Seresht & Jianfei Qiao & Yiying Zhang & Peng Zhao, 2016. "On the skewness of order statistics in multiple-outlier PHR models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(7), pages 817-836, October.
    11. Amarjit Kundu & Shovan Chowdhury & Asok K Nanda, 2014. "Stochastic Comparisons of Parallel Systems of Heterogeneous Generalized Exponential Components," Working papers 162, Indian Institute of Management Kozhikode.
    12. Li, Chen & Li, Xiaohu, 2015. "Likelihood ratio order of sample minimum from heterogeneous Weibull random variables," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 46-53.
    13. Peng Zhao & Yiying Zhang, 2014. "On the maxima of heterogeneous gamma variables with different shape and scale parameters," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(6), pages 811-836, August.
    14. repec:eee:jmvana:v:160:y:2017:i:c:p:31-41 is not listed on IDEAS
    15. Zhao, Peng & Balakrishnan, N., 2009. "Likelihood ratio ordering of convolutions of heterogeneous exponential and geometric random variables," Statistics & Probability Letters, Elsevier, vol. 79(15), pages 1717-1723, August.
    16. Ding, Weiyong & Zhang, Yiying & Zhao, Peng, 2013. "Comparisons of k-out-of-n systems with heterogenous components," Statistics & Probability Letters, Elsevier, vol. 83(2), pages 493-502.

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