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On stochastic orderings between distributions and their sample spacings

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  • Kochar, Subhash C.
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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 X1,X2,...,Xn from a probability distribution with distribution function F. Similarly, let Y1:n[less-than-or-equals, slant]Y2:n[less-than-or-equals, slant]...[less-than-or-equals, slant]Yn:n denote the order statistics of an independent random sample Y1,Y2,...,Yn from G. The corresponding spacings are defined by Ui:n[reverse not equivalent]Xi:n-Xi-1:n and Vi:n[reverse not equivalent]Yi:n-Yi-1:n, for i=1,2,...,n, 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 DFR (decreasing failure rate) distribution, then the vector of Ui:n's is stochastically smaller than the vector of Vi:n's. If instead, we assume that X is smaller than Y in the likelihood ratio order and if either F or G is DFR, then Ui:n is smaller than Vi:n in the hazard rate sense for 1[less-than-or-equals, slant]i[less-than-or-equals, slant]n. Finally, if we make a stronger assumption on the shapes of the distributions that either X or Y has log-convex density, then the random vector of Ui:n's is smaller than the corresponding random vector of Vi:n's in the sense of multivariate likelihood ratio ordering.

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    Bibliographic Info

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

    Volume (Year): 42 (1999)
    Issue (Month): 4 (May)
    Pages: 345-352

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    Handle: RePEc:eee:stapro:v:42:y:1999:i:4:p:345-352

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    Related research

    Keywords: Likelihood ratio ordering Hazard rate ordering Stochastic ordering Multivariate stochastic ordering Multivariate likelihood ratio ordering Dispersive ordering;

    References

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    1. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 467-498, December.
    2. Bartoszewicz, Jaroslaw, 1986. "Dispersive ordering and the total time on test transformation," Statistics & Probability Letters, Elsevier, vol. 4(6), pages 285-288, October.
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    Cited by:
    1. Jongwoo Jeon & Subhash Kochar & Chul Park, 2006. "Dispersive ordering—Some applications and examples," Statistical Papers, Springer, vol. 47(2), pages 227-247, March.
    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. Said, Maher, 2011. "Sequential auctions with randomly arriving buyers," Games and Economic Behavior, Elsevier, vol. 73(1), pages 236-243, September.
    4. 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.
    5. 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.
    6. 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.

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