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Dependence structure of bivariate order statistics with applications to Bayramoglu’s distributions

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  • Huang, J.S.
  • Dou, Xiaoling
  • Kuriki, Satoshi
  • Lin, G.D.

Abstract

We study the dependence structure of bivariate order statistics from bivariate distributions, and prove that if the underlying bivariate distribution H is positive quadrant dependent (PQD) then so is each pair of bivariate order statistics. As an application, we show that if H is PQD, the bivariate distribution K+(n), recently proposed by Bairamov and Bayramoglu (2012) [1], is greater than or equal to Baker’s (2008) [2] distribution H+(n), and hence K+(n) attains a correlation higher than that of H+(n). We give two explicit forms of the intractable K+(n) and prove that for all n≥2, K+(n) is PQD regardless of H. We also show that if H is PQD, K+(n) converges weakly to the Fréchet–Hoeffding upper bound as n tends to infinity.

Suggested Citation

  • Huang, J.S. & Dou, Xiaoling & Kuriki, Satoshi & Lin, G.D., 2013. "Dependence structure of bivariate order statistics with applications to Bayramoglu’s distributions," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 201-208.
    • Handle: RePEc:eee:jmvana:v:114:y:2013:i:c:p:201-208
    DOI: 10.1016/j.jmva.2012.07.009
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    References listed on IDEAS

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    1. H. Barakat, 2001. "The Asymptotic Distribution Theory of Bivariate Order Statistics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(3), pages 487-497, September.
    2. Masaaki Sibuya, 1959. "Bivariate extreme statistics, I," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 11(2), pages 195-210, June.
    3. Baker, Rose, 2008. "An order-statistics-based method for constructing multivariate distributions with fixed marginals," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2312-2327, November.
    4. Boland, Philip J. & Hollander, Myles & Joag-Dev, Kumar & Kochar, Subhash, 1996. "Bivariate Dependence Properties of Order Statistics," Journal of Multivariate Analysis, Elsevier, vol. 56(1), pages 75-89, January.
    5. Lin, G.D. & Huang, J.S., 2010. "A note on the maximum correlation for Baker's bivariate distributions with fixed marginals," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2227-2233, October.
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    Cited by:

    1. Konul Bayramoglu Kavlak, 2017. "Reliability and mean residual life functions of coherent systems in an active redundancy," Naval Research Logistics (NRL), John Wiley & Sons, vol. 64(1), pages 19-28, February.
    2. Gulder Kemalbay & Ismihan Bayramoglu (Bairamov), 2015. "Joint distribution of new sample rank of bivariate order statistics," Journal of Applied Statistics, Taylor & Francis Journals, vol. 42(10), pages 2280-2289, October.
    3. Dou, Xiaoling & Kuriki, Satoshi & Lin, Gwo Dong & Richards, Donald, 2016. "EM algorithms for estimating the Bernstein copula," Computational Statistics & Data Analysis, Elsevier, vol. 93(C), pages 228-245.
    4. Xiaoling Dou & Satoshi Kuriki & Gwo Dong Lin & Donald Richards, 2021. "Dependence Properties of B-Spline Copulas," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 283-311, February.

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