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Mixed systems with minimal and maximal lifetime variances


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  • Marek Beśka


  • Krzysztof Jasiński


  • Tomasz Rychlik


  • Marcin Spryszyński


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    We consider the mixed systems composed of a fixed number of components whose lifetimes are i.i.d. with a known distribution which has a positive and finite variance. We show that a certain of the k-out-of-n systems has the minimal lifetime variance, and the maximal one is attained by a mixture of series and parallel systems. The number of the k-out-of-n system, and the probability weights of the mixture depend on the first two moments of order statistics of the parent distribution of the component lifetimes. We also show methods of calculating extreme system lifetime variances under various restrictions on the system lifetime expectations, and vice versa. Copyright The Author(s) 2012

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    Bibliographic Info

    Article provided by Springer in its journal Metrika.

    Volume (Year): 75 (2012)
    Issue (Month): 7 (October)
    Pages: 877-894

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    Handle: RePEc:spr:metrik:v:75:y:2012:i:7:p:877-894

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    Keywords: Mixed system; k-out-of-n system; Samaniego signature; Order statistic; i.i.d. random variables; Variance;


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    1. Rychlik, Tomasz, 2008. "Extreme variances of order statistics in dependent samples," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1577-1582, September.
    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. Nickos Papadatos, 1997. "A Note on Maximum Variance of Order Statistics from Symmetric Populations," Annals of the Institute of Statistical Mathematics, Springer, vol. 49(1), pages 117-121, March.
    4. Rychlik, Tomasz, 1994. "Distributions and expectations of order statistics for possibly dependent random variables," Journal of Multivariate Analysis, Elsevier, vol. 48(1), pages 31-42, January.
    5. Nickos Papadatos, 1995. "Maximum variance of order statistics," Annals of the Institute of Statistical Mathematics, Springer, vol. 47(1), pages 185-193, January.
    6. Papathanasiou, V., 1990. "Some characterizations of distributions based on order statistics," Statistics & Probability Letters, Elsevier, vol. 9(2), pages 145-147, February.
    7. Balakrishnan, N. & Balasubramanian, K., 1993. "Equivalence of Hartley--David--Gumbel and Papathanasiou bounds and some further remarks," Statistics & Probability Letters, Elsevier, vol. 16(1), pages 39-41, January.
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