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Stochastic orderings between distributions and their sample spacings - II

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  • Khaledi, Baha-Eldin
  • Kochar, Subhash
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    Abstract

    Let X1:n[less-than-or-equals, slant]X2:n[less-than-or-equals, slant]...[less-than-or-equals, slant]Xn:n denote the order statistics of a random sample of size n from a probability distribution with distribution function F. Similarly, let Y1:m[less-than-or-equals, slant]Y2:m[less-than-or-equals, slant]...[less-than-or-equals, slant]Ym:m denote the order statistics of an independent random sample of size m from another distribution with distribution function G. We assume that F and G are absolutely continuous with common support (0,[infinity]). The corresponding normalized spacings are defined by Ui:n[reverse not equivalent](n-i+1)(Xi:n-Xi-1:n) and Vj:m[reverse not equivalent](m-j+1)(Yj:m-Yj-1:m), for i=1,...,n and j=1,...,m, where X0:n=Y0:n[reverse not equivalent]0. It is proved that if X is smaller than Y in the hazard rate order sense and if either F or G is a decreasing failure rate (DFR) distribution, then Ui:n is stochastically smaller than Vj:m for i[less-than-or-equals, slant]j and n-i[greater-or-equal, slanted]m-j. If instead, we assume that X is smaller than Y in the likelihood ratio order and if either F or G is DFR, then this result can be strengthened from stochastic ordering to hazard rate ordering. Finally, under a stronger assumption on the shapes of the distributions that either F or G has log-convex density, it is proved that X being smaller than Y in the likelihood ratio order implies that Ui:n is smaller than Vj:m in the sense of likelihood ratio ordering for i[less-than-or-equals, slant]j and n-i=m-j.

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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 44 (1999)
    Issue (Month): 2 (August)
    Pages: 161-166

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    Handle: RePEc:eee:stapro:v:44:y:1999:i:2:p:161-166

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    Related research

    Keywords: Likelihood ratio ordering Hazard rate ordering Stochastic ordering Dispersive ordering Normalized spacings;

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    Cited by:
    1. Hoppe, Heidrun C. & Moldovanu, Benny & Sela, Aner, 2005. "The Theory of Assortative Matching Based on Costly Signals," Discussion Paper Series of SFB/TR 15 Governance and the Efficiency of Economic Systems 85, Free University of Berlin, Humboldt University of Berlin, University of Bonn, University of Mannheim, University of Munich.
    2. Belzunce, Félix & Mercader, José-Angel & Ruiz, José-María & Spizzichino, Fabio, 2009. "Stochastic comparisons of multivariate mixture models," Journal of Multivariate Analysis, Elsevier, vol. 100(8), pages 1657-1669, September.
    3. 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.
    4. 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.
    5. Jongwoo Jeon & Subhash Kochar & Chul Park, 2006. "Dispersive ordering—Some applications and examples," Statistical Papers, Springer, vol. 47(2), pages 227-247, March.
    6. Hu, Taizhong & Wei, Ying, 2001. "Stochastic comparisons of spacings from restricted families of distributions," Statistics & Probability Letters, Elsevier, vol. 53(1), pages 91-99, May.
    7. Eryilmaz, Serkan, 2012. "On the mean residual life of a k-out-of-n:G system with a single cold standby component," European Journal of Operational Research, Elsevier, vol. 222(2), pages 273-277.

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