Advertisement

Metrika

pp 1–25 | Cite as

Ordering extremes of exponentiated location-scale models with dependent and heterogeneous random samples

  • Sangita Das
  • Suchandan KayalEmail author
Article
  • 73 Downloads

Abstract

This paper is devoted to some ordering results for the largest and the smallest order statistics arising from dependent heterogeneous exponentiated location-scale random observations. We assume that the sets of observations are sharing a common or different Archimedean copula(s). Sufficient conditions for which the usual stochastic order and the reversed hazard rate order between the extreme order statistics hold are derived. Various numerical examples are provided for the illustration of the proposed results. Finally, some applications of the comparison results in engineering reliability and auction theory are presented.

Keywords

Archimedean copula Order statistics Majorization Stochastic orders 

Mathematics Subject Classification

60E15 62G30 60K10 

Notes

Acknowledgements

The authors would like to thank the Editor in Chief Professor Maria Kateri and three anonymous reviewers for their positive remarks and useful comments. One of the authors, Sangita Das thanks the financial support provided by the MHRD, Government of India. Suchandan Kayal gratefully acknowledges the partial financial support for this research work under a Grant MTR/2018/000350, SERB, India.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. Balakrishnan N, Zhao P (2013) Ordering properties of order statistics from heterogeneous populations: a review with an emphasis on some recent developments. Probab Eng Inf Sci 27(4):403–443MathSciNetCrossRefGoogle Scholar
  2. Balakrishnan N, Haidari A, Masoumifard K (2015) Stochastic comparisons of series and parallel systems with generalized exponential components. IEEE Trans Reliab 64(1):333–348CrossRefGoogle Scholar
  3. Balakrishnan N, Nanda P, Kayal S (2018) Ordering of series and parallel systems comprising heterogeneous generalized modified Weibull components. Appl Stoch Models Bus Ind 34(6):816–834MathSciNetCrossRefGoogle Scholar
  4. Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing: probability models. Holt, Rinehart and Winston, New YorkzbMATHGoogle Scholar
  5. Barmalzan G, Payandeh Najafabadi AT, Balakrishnan N (2017) Ordering properties of the smallest and largest claim amounts in a general scale model. Scand Actuar J 2017(2):105–124MathSciNetCrossRefGoogle Scholar
  6. Bartoszewicz J (1986) Dispersive ordering and the total time on test transformation. Stat Probab Lett 4(6):285–288MathSciNetCrossRefGoogle Scholar
  7. Bashkar E, Torabi H, Roozegar R (2017) Stochastic comparisons of extreme order statistics in the heterogeneous exponentiated scale model. J Stat Theory Appl 16(2):219–238MathSciNetGoogle Scholar
  8. Bashkar E, Torabi H, Dolati A, Belzunce F (2018) \(f\)-Majorization with applications to stochastic comparison of extreme order statistics. J Stat Theory Appl 17(3):520–536MathSciNetGoogle Scholar
  9. Boland PJ, Shaked M, Shanthikumar JG (1998) Stochastic ordering of order statistics. In: Balakrishnan N, Rao CR (eds) Handbook of statistics, vol 16. Elsevier, Amsterdam, pp 89–103 zbMATHGoogle Scholar
  10. Boland PJ, Hu T, Shaked M, Shanthikumar JG (2005) Stochastic ordering of order statistics II. In: Modeling uncertainty, vol 46 of International series in operations research & management science. Springer, Boston, MA, pp 607–623Google Scholar
  11. Chowdhury S, Kundu A (2017) Stochastic comparison of parallel systems with log-Lindley distributed components. Oper Res Lett 45(3):199–205MathSciNetCrossRefGoogle Scholar
  12. David H, Nagaraja H (2003) Order statistics, 3rd edn. Wiley, New York CrossRefGoogle Scholar
  13. Ding W, Zhang Y (2018) Relative ageing of series and parallel systems: effects of dependence and heterogeneity among components. Oper Res Lett 46(2):219–224MathSciNetCrossRefGoogle Scholar
  14. Dolati A, Towhidi M, Shekari M (2011) Stochastic and dependence comparisons between extreme order statistics in the case of proportional reversed hazard model. J Iran Stat Soc 10(1):29–43MathSciNetzbMATHGoogle Scholar
  15. Fang R, Li X, Yan R (2015) Impact of dependence among valuations on expected revenue in auctions. Am J Math Manag Sci 34(3):234–264Google Scholar
  16. Fang R, Li C, Li X (2016) Stochastic comparisons on sample extremes of dependent and heterogenous observations. Statistics 50(4):930–955MathSciNetCrossRefGoogle Scholar
  17. Fang R, Li C, Li X (2018) Ordering results on extremes of scaled random variables with dependence and proportional hazards. Statistics 52(2):458–478MathSciNetCrossRefGoogle Scholar
  18. Hazra NK, Kuiti MR, Finkelstein M, Nanda AK (2017) On stochastic comparisons of maximum order statistics from the location-scale family of distributions. J Multivar Anal 160:31–41MathSciNetCrossRefGoogle Scholar
  19. Kayal S (2019) Stochastic comparisons of series and parallel systems with Kumaraswamy-G distributed components. Am J Math Manag Sci 38(1):1–22Google Scholar
  20. Khaledi BE, Kochar SC (2002) Dispersive ordering among linear combinations of uniform random variables. J Stat Plan Inference 100(1):13–21MathSciNetCrossRefGoogle Scholar
  21. Khaledi BE, Farsinezhad S, Kochar SC (2011) Stochastic comparisons of order statistics in the scale model. J Stat Plan Inference 141(1):276–286MathSciNetCrossRefGoogle Scholar
  22. Kundu A, Chowdhury S (2017) Stochastic comparisons of lifetimes of two series and parallel systems with location-scale family distributed components having Archimedean copulas. arXiv preprint arXiv:1710.00769
  23. Kundu A, Chowdhury S, Nanda AK, Hazra NK (2016) Some results on majorization and their applications. J Comput Appl Math 301:161–177MathSciNetCrossRefGoogle Scholar
  24. Li X, Fang R (2015) Ordering properties of order statistics from random variables of Archimedean copulas with applications. J Multivar Anal 133:304–320MathSciNetCrossRefGoogle Scholar
  25. Li C, Li X (2016) Relative ageing of series and parallel systems with statistically independent and heterogeneous component lifetimes. IEEE Trans Reliab 65(2):1014–1021CrossRefGoogle Scholar
  26. Li C, Li X (2019) Hazard rate and reversed hazard rate orders on extremes of heterogeneous and dependent random variables. Stat Probab Lett 146:104–111MathSciNetCrossRefGoogle Scholar
  27. Li C, Fang R, Li X (2016) Stochastic somparisons of order statistics from scaled and interdependent random variables. Metrika 79(5):553–578MathSciNetCrossRefGoogle Scholar
  28. Marshall AW, Olkin I, Arnold BC (2011) Inequality: theory of majorization and its applications. Springer series in statistics. Springer, New YorkCrossRefGoogle Scholar
  29. McNeil AJ, Nešlehová J (2009) Multivariate archimedean copulas, \(d\)-monotone functions and \(l^1\)-norm symmetric distributions. Ann Stat 37(5B):3059–3097CrossRefGoogle Scholar
  30. Mesfioui M, Kayid M, Izadkhah S (2017) Stochastic comparisons of order statistics from heterogeneous random variables with Archimedean copula. Metrika 80(6–8):749–766MathSciNetCrossRefGoogle Scholar
  31. Nadeba H, Torabi H (2018) Stochastic comparisons of series systems with independent heterogeneous Lomax-exponential components. J Stat Theory Pract 12(4):794–812MathSciNetCrossRefGoogle Scholar
  32. Navarro J, Spizzichino F (2010) Comparisons of series and parallel systems with components sharing the same copula. Appl Stoch Models Bus Ind 26(6):775–791MathSciNetCrossRefGoogle Scholar
  33. Navarro J, Torrado N, del Águila Y (2018) Comparisons between largest order statistics from multiple-outlier models with dependence. Methodol Comput Appl Probab 20(1):411–433MathSciNetCrossRefGoogle Scholar
  34. Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
  35. Pledger G, Proschan F (1971) Comparisons of order statistics and of spacings from heterogeneous distributions. In: Rustagi JS (ed) Optimizing methods in statistics. Academic Press, New York, pp 89–113 zbMATHGoogle Scholar
  36. Proschan F, Sethuraman J (1976) Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability. J Multivar Anal 6(4):608–616MathSciNetCrossRefGoogle Scholar
  37. Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, BerlinCrossRefGoogle Scholar
  38. Singh H, Vijayasree G (1991) Preservation of partial orderings under the formation of k-out-of-n: G systems of iid components. IEEE Trans Reliab 40(3):273–276CrossRefGoogle Scholar
  39. Torrado N (2015) On magnitude orderings between smallest order statistics from heterogeneous beta distributions. J Math Anal Appl 426(2):824–838MathSciNetCrossRefGoogle Scholar
  40. Torrado N (2017) Stochastic comparisons between extreme order statistics from scale models. Statistics 51(6):1359–1376MathSciNetCrossRefGoogle Scholar
  41. Zhang Y, Cai X, Zhao P, Wang H (2019) Stochastic comparisons of parallel and series systems with heterogeneous resilience-scaled components. Statistics 53(1):126–147MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Technology RourkelaRourkelaIndia

Personalised recommendations