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Mixed systems with minimal and maximal lifetime variances

Author

Listed:
  • Marek Beśka
  • Krzysztof Jasiński
  • Tomasz Rychlik
  • Marcin Spryszyński

Abstract

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

Suggested Citation

  • Marek Beśka & Krzysztof Jasiński & Tomasz Rychlik & Marcin Spryszyński, 2012. "Mixed systems with minimal and maximal lifetime variances," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(7), pages 877-894, October.
    • Handle: RePEc:spr:metrik:v:75:y:2012:i:7:p:877-894
    DOI: 10.1007/s00184-011-0357-5
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    References listed on IDEAS

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    1. Nickos Papadatos, 1995. "Maximum variance of order statistics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(1), pages 185-193, January.
    2. Shaked, Moshe & Suarez-Llorens, Alfonso, 2003. "On the Comparison of Reliability Experiments Based on the Convolution Order," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 693-702, January.
    3. 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.
    4. Papathanasiou, V., 1990. "Some characterizations of distributions based on order statistics," Statistics & Probability Letters, Elsevier, vol. 9(2), pages 145-147, February.
    5. 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.
    6. 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.
    7. Rychlik, Tomasz, 2008. "Extreme variances of order statistics in dependent samples," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1577-1582, September.
    8. Nickos Papadatos, 1997. "A Note on Maximum Variance of Order Statistics from Symmetric Populations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 49(1), pages 117-121, March.
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