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Stochastic comparisons of distorted distributions, coherent systems and mixtures with ordered components

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  • Jorge Navarro

    (Universidad de Murcia)

  • Yolanda Águila

    (Universidad de Almería)

Abstract

A distribution function F is a generalized distorted distribution of the distribution functions $$F_1,\ldots ,F_n$$ F 1 , … , F n if $$F=Q(F_1,\ldots ,F_n)$$ F = Q ( F 1 , … , F n ) for an increasing continuous distortion function Q such that $$Q(0,\ldots ,0)=0$$ Q ( 0 , … , 0 ) = 0 and $$Q(1,\ldots ,1)=1$$ Q ( 1 , … , 1 ) = 1 . In this paper, necessary and sufficient conditions for the stochastic (ST) and the hazard rate (HR) orderings of generalized distorted distributions are provided when the distributions $$F_1,\ldots ,F_n$$ F 1 , … , F n are ordered. These results are used to obtain distribution-free ordering properties for coherent systems with heterogeneous components. In particular, we determine all the ST and HR orderings for coherent systems with 1–3 independent components. We also compare systems with dependent components. The results on distorted distributions are also used to get comparisons of finite mixtures.

Suggested Citation

  • Jorge Navarro & Yolanda Águila, 2017. "Stochastic comparisons of distorted distributions, coherent systems and mixtures with ordered components," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 80(6), pages 627-648, November.
    • Handle: RePEc:spr:metrik:v:80:y:2017:i:6:d:10.1007_s00184-017-0619-y
    DOI: 10.1007/s00184-017-0619-y
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    References listed on IDEAS

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    1. S. Goli & M. Asadi, 2017. "A study on the conditional inactivity time of coherent systems," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 80(2), pages 227-241, February.
    2. P. Samadi & M. Rezaei & M. Chahkandi, 2017. "On the residual lifetime of coherent systems with heterogeneous components," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 80(1), pages 69-82, January.
    3. Jorge Navarro, 2016. "Stochastic comparisons of generalized mixtures and coherent systems," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 150-169, March.
    4. Navarro, Jorge & Rychlik, Tomasz, 2010. "Comparisons and bounds for expected lifetimes of reliability systems," European Journal of Operational Research, Elsevier, vol. 207(1), pages 309-317, November.
    5. Zhengcheng Zhang & N. Balakrishnan, 2016. "Representations of the inactivity time for coherent systems with heterogeneous components and some ordered properties," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(1), pages 113-126, January.
    6. Francisco J. Samaniego, 2007. "System Signatures and their Applications in Engineering Reliability," International Series in Operations Research and Management Science, Springer, number 978-0-387-71797-5, December.
    7. Jorge Navarro & Yolanda Águila & Miguel A. Sordo & Alfonso Suárez-Llorens, 2016. "Preservation of Stochastic Orders under the Formation of Generalized Distorted Distributions. Applications to Coherent Systems," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 529-545, June.
    8. Jorge Navarro & M. Carmen Gomis, 2016. "Comparisons in the mean residual life order of coherent systems with identically distributed components," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 32(1), pages 33-47, January.
    9. Navarro, Jorge & Pellerey, Franco & Di Crescenzo, Antonio, 2015. "Orderings of coherent systems with randomized dependent components," European Journal of Operational Research, Elsevier, vol. 240(1), pages 127-139.
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    Cited by:

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    2. Mansour Shrahili & Mohamed Kayid & Mhamed Mesfioui, 2023. "Relative Orderings of Modified Proportional Hazard Rate and Modified Proportional Reversed Hazard Rate Models," Mathematics, MDPI, vol. 11(22), pages 1-28, November.
    3. Nil Kamal Hazra & Maxim Finkelstein, 2018. "On stochastic comparisons of finite mixtures for some semiparametric families of distributions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(4), pages 988-1006, December.
    4. Omid Shojaee & Majid Asadi & Maxim Finkelstein, 2021. "On Some Properties of $$\alpha $$ α -Mixtures," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(8), pages 1213-1240, November.
    5. M. Salehi & Z. Shishebor & M. Asadi, 2019. "On the reliability modeling of weighted k-out-of-n systems with randomly chosen components," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(5), pages 589-605, July.
    6. Ji Hwan Cha & Maxim Finkelstein, 2020. "Is perfect repair always perfect?," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(1), pages 90-104, March.
    7. Elham Khaleghpanah Noughabi & Majid Chahkandi & Majid Rezaei, 2022. "On the Mean and Variance Residual Life Comparisons of Coherent Systems with Identically Distributed Components," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2801-2822, December.
    8. Omid Shojaee & Manoochehr Babanezhad, 2023. "On some stochastic comparisons of arithmetic and geometric mixture models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(5), pages 499-515, July.
    9. Torrado, Nuria & Arriaza, Antonio & Navarro, Jorge, 2021. "A study on multi-level redundancy allocation in coherent systems formed by modules," Reliability Engineering and System Safety, Elsevier, vol. 213(C).

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