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

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Abstract

A distribution function F is a generalized distorted distribution of the distribution functions \(F_1,\ldots ,F_n\) if \(F=Q(F_1,\ldots ,F_n)\) for an increasing continuous distortion function Q such that \(Q(0,\ldots ,0)=0\) and \(Q(1,\ldots ,1)=1\). 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 \(F_1,\ldots ,F_n\) 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.

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References

  • Balakrishnan N, Torrado N (2016) Comparisons between largest order statistics from multiple-outlier models. Statistics 50:176–189

    Article  MATH  MathSciNet  Google Scholar 

  • Barlow RE, Proschan F (1975) Statistical Theory of Reliability and Life Testing. International Series in Decision Processes. Holt, Rinehart and Winston Inc, New York

    MATH  Google Scholar 

  • Belzunce F, Martinez-Riquelme C, Mulero J (2016) An Introduction to Stochastic Orders. Elsevier, Amsterdam

  • Durante F, Sempi C (2016) Principles of Copula Theory. CRC/Chapman & Hall, London

    MATH  Google Scholar 

  • Fernández-Ponce JM, Pellerey F, Rodríguez-Griñolo MR (2016) Some stochastic properties of conditionally dependent frailty models. Statistics 50:649–666

    Article  MATH  MathSciNet  Google Scholar 

  • Goli S, Asadi M (2017) A study on the conditional inactivity time of coherent systems. Metrika 80:227–241

    Article  MATH  MathSciNet  Google Scholar 

  • Hürlimann W (2004) Distortion risk measures and economic capital. N Am Actuar J 8:86–95

    Article  MATH  MathSciNet  Google Scholar 

  • Kochar S, Mukerjee H, Samaniego FJ (1999) The “signature” of a coherent system and its application to comparison among systems. Naval Res Log 46:507–523

    Article  MATH  MathSciNet  Google Scholar 

  • Navarro J (2016) Stochastic comparisons of generalized mixtures and coherent systems. Test 25:150–169

    Article  MATH  MathSciNet  Google Scholar 

  • Navarro J, Shaked M (2006) Hazard rate ordering of order statistics and systems. J Appl Prob 43:391–408

    Article  MATH  MathSciNet  Google Scholar 

  • Navarro J, Rychlik T (2010) Comparisons and bounds for expected lifetimes of reliability systems. Eur J Oper Res 207:309–317

    Article  MATH  MathSciNet  Google Scholar 

  • Navarro J, Rubio R (2011) A note on necessary and sufficient conditions for ordering properties of coherent systems with exchangeable components. Naval Res Log 58:478–489

    Article  MATH  MathSciNet  Google Scholar 

  • Navarro J, Rubio R (2012) Comparisons of coherent systems with non-identically distributed components. J Stat Plan Inf 142:1310–1319

    Article  MATH  MathSciNet  Google Scholar 

  • Navarro J, Gomis MC (2016) Comparisons in the mean residual life order of coherent systems with identically distributed components. Appl Stoch Mod Bus Ind 32:33–47

    Article  MathSciNet  Google Scholar 

  • Navarro J, Samaniego FJ, Balakrishnan N, Bhattacharya D (2008) Applications and extensions of system signatures in engineering reliability. Naval Res Log 55:313–327

    Article  MATH  MathSciNet  Google Scholar 

  • Navarro J, Guillamon A, Ruiz MC (2009) Generalized mixtures in reliability modelling: applications to the construction of bathtub shaped hazard models and the study of systems. Appl Stoch Mod Bus Ind 25:323–337

    Article  MATH  MathSciNet  Google Scholar 

  • Navarro J, del Aguila Y, Sordo MA, Suárez-Llorens A (2013) Stochastic ordering properties for systems with dependent identically distributed components. Appl Stoch Mod Bus Ind 29:264–278

    Article  MATH  MathSciNet  Google Scholar 

  • Navarro J, del Aguila Y, Sordo MA, Suárez-Llorens A (2014) Preservation of reliability classes under the formation of coherent systems. Appl Stoch Mod Bus Ind 30:444–454

    Article  MathSciNet  Google Scholar 

  • Navarro J, Pellerey F, Di Crescenzo A (2015) Orderings of coherent systems with randomized dependent components. Eur J Oper Res 240:127–139

    Article  MATH  MathSciNet  Google Scholar 

  • Navarro J, del Aguila Y, Sordo MA, Suárez-Llorens A (2016) Preservation of stochastic orders under the formation of generalized distorted distributions. Applications to coherent systems. Meth Comp Appl Prob 18:529–545

    Article  MATH  MathSciNet  Google Scholar 

  • Nelsen RB (2006) An Introduction to Copulas. Springer Series in Statistics, 2nd edn. Springer, New York

    Google Scholar 

  • Samadi P, Rezaei M, Chahkandi M (2017) On the residual lifetime of coherent systems with heterogeneous components. Metrika 80:69–82

    Article  MATH  MathSciNet  Google Scholar 

  • Samaniego FJ (2007) System Signatures and their Applications in Engineering Reliability. Springer, New York

    Book  MATH  Google Scholar 

  • Samaniego FJ, Navarro J (2016) On comparing coherent systems with heterogeneous components. Adv Appl Prob 48:88–111

    Article  MATH  MathSciNet  Google Scholar 

  • Shaked M, Shanthikumar JG (2007) Stochastic Orders. Springer, New York

    Book  MATH  Google Scholar 

  • Zhang Z, Balakrishnan N (2016) Representations of the inactivity time for coherent systems with heterogeneous components and some ordered properties. Metrika 79:113–126

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

We would like to thank the anonymous reviewers for several helpful suggestions that allow us to improve the paper. JN was supported in part by Ministerio de Economía y Competitividad of Spain under Grant MTM2012-34023-FEDER.

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Authors and Affiliations

  1. Facultad de Matemáticas, Universidad de Murcia, 30100, Murcia, Spain

    Jorge Navarro

  2. Departamento de Matemáticas, Universidad de Almería, Almería, Spain

    Yolanda del Águila

Authors
  1. Jorge Navarro
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  2. Yolanda del Águila
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Correspondence to Jorge Navarro.

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Navarro, J., del Águila, Y. Stochastic comparisons of distorted distributions, coherent systems and mixtures with ordered components. Metrika 80, 627–648 (2017). https://doi.org/10.1007/s00184-017-0619-y

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  • DOI: https://doi.org/10.1007/s00184-017-0619-y

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