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Bivariate Dependence Properties of Order Statistics

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  • Boland, Philip J.
  • Hollander, Myles
  • Joag-Dev, Kumar
  • Kochar, Subhash
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    Abstract

    IfX1, ...,Xnare random variables we denote byX(1)[less-than-or-equals, slant]X(2)[less-than-or-equals, slant]...[less-than-or-equals, slant]X(n)their respective order statistics. In the case where the random variables are independent and identically distributed, one may demonstrate very strong notions of dependence between any two order statisticsX(i)andX(j). If in particular the random variables are independent with a common density or mass function, thenX(i)andX(j)areTP2dependent for anyiandj. In this paper we consider the situation in which the random variablesX1, ...,Xnare independent but otherwise arbitrarily distributed. We show that for anyi t|X(i)>s] is an increasing function ofs. This is a stronger form of dependence betweenX(i)andX(j)than that of association, but we also show that among the hierarchy of notions of bivariate dependence this is the strongest possible under these circumstances. It is also shown that in this situation,P[X(j)>t|X(i)>s] is a decreasing function ofi=1, ...,nfor any fixeds

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 56 (1996)
    Issue (Month): 1 (January)
    Pages: 75-89

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    Handle: RePEc:eee:jmvana:v:56:y:1996:i:1:p:75-89

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    Related research

    Keywords: order statistics associated random variables right corner set increasing right tail increasing stochastically increasing TP2property positively quadrant dependent reliability counting processes exchangeability finite sampling problem kout ofnsystem (null);

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    Cited by:
    1. Franco, Manuel & Vivo, Juana-María, 2010. "A multivariate extension of Sarhan and Balakrishnan's bivariate distribution and its ageing and dependence properties," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 491-499, March.
    2. Navarro, Jorge & Balakrishnan, N., 2010. "Study of some measures of dependence between order statistics and systems," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 52-67, January.
    3. Montes, Ignacio & Miranda, Enrique & Montes, Susana, 2014. "Decision making with imprecise probabilities and utilities by means of statistical preference and stochastic dominance," European Journal of Operational Research, Elsevier, vol. 234(1), pages 209-220.
    4. Avérous, Jean & Genest, Christian & C. Kochar, Subhash, 2005. "On the dependence structure of order statistics," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 159-171, May.
    5. Fountain, Robert L. & Herman Jr., John R. & Rustvold, D. Leif, 2008. "An application of Kendall distributions and alternative dependence measures: SPX vs. VIX," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 469-472, April.
    6. Hu, Taizhong & Xie, Chaode, 2006. "Negative dependence in the balls and bins experiment with applications to order statistics," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1342-1354, July.
    7. Zhuang, Weiwei & Yao, Junchao & Hu, Taizhong, 2010. "Conditional ordering of order statistics," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 640-644, March.
    8. Mao, Tiantian & Hu, Taizhong, 2010. "Stochastic properties of INID progressive Type-II censored order statistics," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1493-1500, July.
    9. Khaledi, Baha-Eldin & Kochar, Subhash, 2000. "Stochastic Comparisons and Dependence among Concomitants of Order Statistics," Journal of Multivariate Analysis, Elsevier, vol. 73(2), pages 262-281, May.
    10. Dolati, Ali & Genest, Christian & Kochar, Subhash C., 2008. "On the dependence between the extreme order statistics in the proportional hazards model," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 777-786, May.

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