IDEAS home Printed from https://ideas.repec.org/a/spr/metrik/v79y2016i6d10.1007_s00184-015-0570-8.html
   My bibliography  Save this article

Evaluations of expectations of order statistics and spacings based on IFR distributions

Author

Listed:
  • Agnieszka Goroncy

    (Nicolaus Copernicus University)

  • Tomasz Rychlik

    (Polish Academy of Sciences)

Abstract

We consider i.i.d. random variables $$X_1,\ldots , X_n$$ X 1 , … , X n with a distribution function F preceding the exponential distribution function V in the convex transform order which means that F has an increasing failure rate. We determine sharp upper bounds on the expectations of order statistics and spacings based on $$X_1,\ldots , X_n$$ X 1 , … , X n , expressed in the population standard deviation units. We also specify the distributions for which all these bounds are attained. Finally, we indicate some reliability applications.

Suggested Citation

  • Agnieszka Goroncy & Tomasz Rychlik, 2016. "Evaluations of expectations of order statistics and spacings based on IFR distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(6), pages 635-657, August.
    • Handle: RePEc:spr:metrik:v:79:y:2016:i:6:d:10.1007_s00184-015-0570-8
    DOI: 10.1007/s00184-015-0570-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00184-015-0570-8
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00184-015-0570-8?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Agnieszka Goroncy & Tomasz Rychlik, 2015. "Optimal bounds on expectations of order statistics and spacings from nonparametric families of distributions generated by convex transform order," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(2), pages 175-204, February.
    2. Gajek, Leslaw & Rychlik, Tomasz, 1996. "Projection Method for Moment Bounds on Order Statistics from Restricted Families: I. Dependent Case," Journal of Multivariate Analysis, Elsevier, vol. 57(1), pages 156-174, April.
    3. Gajek, Leslaw & Rychlik, Tomasz, 1998. "Projection Method for Moment Bounds on Order Statistics from Restricted Families, : II. Independent Case," Journal of Multivariate Analysis, Elsevier, vol. 64(2), pages 156-182, February.
    4. Danielak, Katarzyna & Rychlik, Tomasz, 2003. "Sharp bounds for expectations of spacings from DDA and DFRA families," Statistics & Probability Letters, Elsevier, vol. 65(4), pages 303-316, December.
    5. Tomasz Rychlik, 2014. "Maximal Expectations of Extreme Order Statistics from Increasing Density and Failure Rate Populations," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 43(10-12), pages 2199-2213, May.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Rychlik, Tomasz, 2017. "Evaluations of quantiles of system lifetime distributions," European Journal of Operational Research, Elsevier, vol. 256(3), pages 935-944.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Agnieszka Goroncy & Tomasz Rychlik, 2015. "Optimal bounds on expectations of order statistics and spacings from nonparametric families of distributions generated by convex transform order," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(2), pages 175-204, February.
    2. Danielak, Katarzyna & Rychlik, Tomasz, 2003. "Sharp bounds for expectations of spacings from DDA and DFRA families," Statistics & Probability Letters, Elsevier, vol. 65(4), pages 303-316, December.
    3. Paweł Marcin Kozyra & Tomasz Rychlik, 2017. "Lower and upper bounds on the variances of spacings," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(2), pages 417-428, April.
    4. Tomasz Rychlik, 2014. "Non-positive upper bounds on expectations of small order statistics from DDA and DFRA populations," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(4), pages 539-557, May.
    5. Tomasz Rychlik, 2001. "Stability of Order Statistics under Dependence," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(4), pages 877-894, December.
    6. Danielak, Katarzyna & Raqab, Mohammad Z., 2004. "Sharp bounds on expectations of kth record spacings from restricted families," Statistics & Probability Letters, Elsevier, vol. 69(2), pages 175-187, August.
    7. Rychlik, Tomasz, 2002. "Best upper quantile evaluations for NWU distributions," Statistics & Probability Letters, Elsevier, vol. 58(2), pages 175-184, June.
    8. Rychlik, Tomasz, 2009. "Tight evaluations for expectations of small order statistics from symmetric and symmetric unimodal populations," Statistics & Probability Letters, Elsevier, vol. 79(12), pages 1488-1493, June.
    9. Mariusz Bieniek, 2015. "Optimal evaluations for the bias of trimmed means of $$k$$ k th record values," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(4), pages 437-460, May.
    10. Mariusz Bieniek & Agnieszka Goroncy, 2020. "Sharp lower bounds on expectations of gOS based on DGFR distributions," Statistical Papers, Springer, vol. 61(3), pages 1027-1042, June.
    11. Rychlik, Tomasz, 2017. "Evaluations of quantiles of system lifetime distributions," European Journal of Operational Research, Elsevier, vol. 256(3), pages 935-944.
    12. Okolewski, Andrzej, 2017. "Extremal properties of order statistic distributions for dependent samples with partially known multidimensional marginals," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 1-9.
    13. Gajek, Leslaw & Rychlik, Tomasz, 1998. "Projection Method for Moment Bounds on Order Statistics from Restricted Families, : II. Independent Case," Journal of Multivariate Analysis, Elsevier, vol. 64(2), pages 156-182, February.
    14. Okolewski, Andrzej & Rychlik, Tomasz, 2001. "Sharp distribution-free bounds on the bias in estimating quantiles via order statistics," Statistics & Probability Letters, Elsevier, vol. 52(2), pages 207-213, April.
    15. J. M. Fernández-Ponce & M. R. Rodríguez-Griñolo, 2017. "New properties of the orthant convex-type stochastic orders," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(3), pages 618-637, September.
    16. Tomasz Rychlik, 2009. "Bounds on expectations of small order statistics from decreasing density populations," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 70(3), pages 369-381, November.
    17. Mariusz Bieniek, 2007. "Variation diminishing property of densities of uniform generalized order statistics," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 65(3), pages 297-309, May.
    18. Mariusz Bieniek & Luiza Pańczyk, 2023. "On the choice of the optimal single order statistic in quantile estimation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(2), pages 303-333, April.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:metrik:v:79:y:2016:i:6:d:10.1007_s00184-015-0570-8. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.