Skip to main content
SpringerLink
Log in

HR and RHR orderings of generalized order statistics

  • Published:
Metrika Aims and scope Submit manuscript

Abstract

In this article, we compare generalized order statistics in terms of increasing convex, mean residual life, hazard rate and reversed hazard rate orderings and establish some general results, including resolving an open problem. Some counterexamples are also provided for the cases wherein the results are not valid in general. Finally, some new applications of these results are demonstrated for sequential systems, k-record values and Pfeifer’s records.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Ahmadi J, Arghami NR (2001) Some univariate stochastic orders on record values. Commun Stat Theor Meth. 30(1):69–74

    Article  MathSciNet  MATH  Google Scholar 

  • Alimohammadi M, Alamatsaz MH (2011) Some new results on unimodality of generalized order statistics and their spacings. Stat Probab Lett 81(11):1677–1682

    Article  MathSciNet  MATH  Google Scholar 

  • Alimohammadi M, Esna-Ashari M, Cramer E (2021) On dispersive and star orderings of random variables and order statistics. Stat Probab Lett 170:109014

    Article  MathSciNet  MATH  Google Scholar 

  • Arnold BC, Balakrishnan N, Nagaraja HN (1992) A First course in order statistics. John Wiley & Sons, New York

    MATH  Google Scholar 

  • Arnold BC, Balakrishnan N, Nagaraja HN (1998) Records. John Wiley & Sons, New York

    Book  MATH  Google Scholar 

  • Balakrishnan N, Belzunce F, Sordo MA, Suárez-Llorens A (2012) Increasing directionally convex orderings of random vectors having the same copula, and their use in comparing ordered data. J Multivar Anal 105(1):45–54

    Article  MathSciNet  MATH  Google Scholar 

  • Belzunce F, Mercader JA, Ruiz JM (2003) Multivariate aging properties of epoch times of nonhomogeneous processes. J Multivar Anal 84:335–350

    Article  MathSciNet  MATH  Google Scholar 

  • Belzunce F, Mercader JA, Ruiz JM (2005) Stochastic comparisons of generalized order statistics. Probab Eng Inf Sci 19(1):99–120

    Article  MathSciNet  MATH  Google Scholar 

  • Belzunce F, Martínez-Riquelme C, Ruiz JM (2013) On sufficient conditions for mean residual life and related orders. Comput Stat Data Anal 61:199–210

    Article  MathSciNet  MATH  Google Scholar 

  • Burkschat M, Cramer E, Kamps U (2003) Dual generalized order statistics. Metron 61(1):13–26

    MathSciNet  MATH  Google Scholar 

  • Burkschat M, Navarro J (2018) Stochastic comparisons of systems based on sequential order statistics via properties of distorted distributions. Probab Eng Inf Sci 32(2):246–274

    Article  MathSciNet  MATH  Google Scholar 

  • Burkschat M, Torrado N (2014) On the reversed hazard rate of sequential order statistics. Stat Probab Lett 85:106–113

    Article  MathSciNet  MATH  Google Scholar 

  • Cramer E, Kamps U (1996) Sequential order statistics and k-out-of-n systems with sequentially adjusted failure rates. Ann Inst Stat Math 48(3):535–549

    Article  MathSciNet  MATH  Google Scholar 

  • Cramer E, Kamps U (2001) Sequential \(k\)-out-of-\(n\) systems. In: Balakrishnan N, Rao CR (eds) Handbook of statistics: advances in reliability, vol 20. Elsevier, Amsterdam, pp 301–372

    MATH  Google Scholar 

  • Cramer E, Kamps U (2003) Marginal distributions of sequential and generalized order statistics. Metrika 58(3):293–310

    Article  MathSciNet  MATH  Google Scholar 

  • Cramer E, Kamps U, Rychlik T (2004) Unimodality of uniform generalized order statistics, with applications to mean bounds. Ann Inst Stat Math 56(1):183–192

    Article  MathSciNet  MATH  Google Scholar 

  • David HA, Nagaraja HN (2003) Order statistics, 3rd edn. John Wiley & Sons, Hoboken, New Jersey

    Book  MATH  Google Scholar 

  • Esna-Ashari M, Alimohammadi M, Cramer E (2020) Some new results on likelihood ratio ordering and aging properties of generalized order statistics. Commun Stat Theor Meth 1–25. https://doi.org/10.1080/03610926.2020.1818103

  • Franco M, Ruiz JM, Ruiz MC (2002) Stochastic orderings between spacings of generalized order statistics. Probab Eng Inf Sci 16(4):471–484

    Article  MathSciNet  MATH  Google Scholar 

  • Hu T, Zhuang W (2005) A note on stochastic comparisons of generalized order statistics. Stat Prob Lett 72(2):163–170

    Article  MathSciNet  MATH  Google Scholar 

  • Kamps U (1995) A concept of generalized order statistics. J Stat Plann Infer 48(1):1–23

    Article  MathSciNet  MATH  Google Scholar 

  • Khaledi BE (2005) Some new results on stochastic orderings between generalized order statistics. J Iranian Stat Soc 4(1):35–49

    MathSciNet  MATH  Google Scholar 

  • Kochar SC (1990) Some partial ordering results on record values. Commun Stat Theor Meth 19(1):299–306

    Article  MathSciNet  MATH  Google Scholar 

  • Navarro J, Burkschat M (2011) Coherent systems based on sequential order statistics. Nav Res Logist 58(2):123–135

    Article  MathSciNet  MATH  Google Scholar 

  • Raqab MZ, Amin WA (1997) A note on reliability properties of k-record statistics. Metrika 46(1):245–251

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  • Torrado N, Lillo RE, Wiper MP (2012) Sequential order statistics: ageing and stochastic orderings. Meth Comp Appl Prob 14(3):579–596

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao P, Balakrishnan N (2009) Stochastic comparisons and properties of conditional generalized order statistics. J Stat Plan Infer 139(9):2920–2932

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We express our sincere thanks to the anonymous reviewers and the Associate Editor for their incisive comments on an earlier version of this manuscript which led to this improved version.

Author information

Authors and Affiliations

  1. Insurance Research Center, Tehran, Iran

    Maryam Esna-Ashari

  2. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada

    Narayanaswamy Balakrishnan

  3. Department of Statistics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran

    Mahdi Alimohammadi

Authors
  1. Maryam Esna-Ashari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Narayanaswamy Balakrishnan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mahdi Alimohammadi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maryam Esna-Ashari.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Esna-Ashari, M., Balakrishnan, N. & Alimohammadi, M. HR and RHR orderings of generalized order statistics. Metrika 86, 131–148 (2023). https://doi.org/10.1007/s00184-022-00865-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00184-022-00865-2

Keywords

Search

Navigation