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Distribution and expectation bounds on order statistics from possibly dependent variates

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  • Papadatos, Nickos

Abstract

Let X1,X2,...,Xn be n random variables with an arbitrary n-variate distribution. We say that the X's are maximally (resp. minimally) stable of order j (j[set membership, variant]{1,2,...,n}), if the distribution F(j) of max{Xk1,...,Xkj} (resp. G(j) of min{Xk1,...,Xkj}) is the same, for any j-subset {k1,...,kj} of {1,2,...,n}. Under the assumption of maximal (resp. minimal) stability of order j, sharp upper (resp. lower) bounds are given for the distribution Fk:n of the kth order statistic Xk:n, in terms of F(j) (resp. G(j)), and the corresponding expectation bounds are derived. Moreover, some expectation bounds in the case of j-independent-F samples (i.e., when each j-tuple Xk1,...,Xkj is independent with a common marginal distribution F) are given.

Suggested Citation

  • Papadatos, Nickos, 2001. "Distribution and expectation bounds on order statistics from possibly dependent variates," Statistics & Probability Letters, Elsevier, vol. 54(1), pages 21-31, August.
    • Handle: RePEc:eee:stapro:v:54:y:2001:i:1:p:21-31
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    References listed on IDEAS

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    1. Rychlik, Tomasz, 1992. "Stochastically extremal distributions of order statistics for dependent samples," Statistics & Probability Letters, Elsevier, vol. 13(5), pages 337-341, April.
    2. Caraux, G. & Gascuel, O., 1992. "Bounds on distribution functions of order statistics for dependent variates," Statistics & Probability Letters, Elsevier, vol. 14(2), pages 103-105, May.
    3. Gascuel, O. & Caraux, G., 1992. "Bounds on expectations of order statistics via extremal dependences," Statistics & Probability Letters, Elsevier, vol. 15(2), pages 143-148, September.
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    Cited by:

    1. Okolewski, Andrzej, 2017. "Extremal properties of order statistic distributions for dependent samples with partially known multidimensional marginals," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 1-9.
    2. Navarro, Jorge & Ruiz, Jose M. & Sandoval, Carlos J., 2005. "A note on comparisons among coherent systems with dependent components using signatures," Statistics & Probability Letters, Elsevier, vol. 72(2), pages 179-185, April.
    3. Navarro, Jorge & Rychlik, Tomasz, 2007. "Reliability and expectation bounds for coherent systems with exchangeable components," Journal of Multivariate Analysis, Elsevier, vol. 98(1), pages 102-113, January.

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