IDEAS home Printed from https://ideas.repec.org/a/spr/testjl/v32y2023i2d10.1007_s11749-022-00844-9.html
   My bibliography  Save this article

Sharp inequalities of Bienaymé–Chebyshev and Gauß type for possibly asymmetric intervals around the mean

Author

Listed:
  • Roxana A. Ion

    (ASML Veldhoven)

  • Chris A. J. Klaassen

    (University of Amsterdam)

  • Edwin R. van den Heuvel

    (Eindhoven University of Technology)

Abstract

Sharp upper bounds are proved for the probability that a standardized random variable takes on a value outside a possibly asymmetric interval around 0. Six classes of distributions for the random variable are considered, namely the general class of ‘distributions’, the class of ‘symmetric distributions’, of ‘concave distributions’, of ‘unimodal distributions’, of ‘unimodal distributions with coinciding mode and mean’, and of ‘symmetric unimodal distributions’. In this way, results by Gauß (Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores 5:1–58, 1823), Bienaymé (C R Hebd Séance Acad Sci Paris 37:309–24, 1853), Bienaymé (C R Hebd Séance Acad Sci Paris 37:309–24, 1853), Chebyshev (Journal de mathématiques pures et appliqués (2) 12:177–184, 1867), and Cantelli (Atti del Congresso Internazionale dei Matematici 6:47–59, 1928) are generalized. For some of the known inequalities, such as the Gauß inequality, an alternative proof is given.

Suggested Citation

  • Roxana A. Ion & Chris A. J. Klaassen & Edwin R. van den Heuvel, 2023. "Sharp inequalities of Bienaymé–Chebyshev and Gauß type for possibly asymmetric intervals around the mean," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(2), pages 566-601, June.
    • Handle: RePEc:spr:testjl:v:32:y:2023:i:2:d:10.1007_s11749-022-00844-9
    DOI: 10.1007/s11749-022-00844-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11749-022-00844-9
    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/s11749-022-00844-9?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. Haruhiko Ogasawara, 2019. "The multiple Cantelli inequalities," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 28(3), pages 495-506, September.
    2. Bhat, M. Ashraf & Kosuru, G. Sankara Raju, 2022. "Generalizations of some concentration inequalities," Statistics & Probability Letters, Elsevier, vol. 182(C).
    3. Jonathan Rougier & Michael Goldstein & Leanna House, 2013. "Second-Order Exchangeability Analysis for Multimodel Ensembles," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(503), pages 852-863, September.
    4. Haruhiko Ogasawara, 2020. "The multivariate Markov and multiple Chebyshev inequalities," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 49(2), pages 441-453, January.
    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. Mengshuo Zhao & Narayanaswamy Balakrishnan & Chuancun Yin, 2024. "Extremal cases of distortion risk measures with partial information," Papers 2404.13637, arXiv.org, revised Oct 2024.

    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. Bhat, M. Ashraf & Kosuru, G. Sankara Raju, 2022. "Generalizations of some concentration inequalities," Statistics & Probability Letters, Elsevier, vol. 182(C).
    2. Esther Salazar & Dorit Hammerling & Xia Wang & Bruno Sansó & Andrew O. Finley & Linda O. Mearns, 2016. "Observation-based blended projections from ensembles of regional climate models," Climatic Change, Springer, vol. 138(1), pages 55-69, September.
    3. Beghin, Luisa & Macci, Claudio & Ricciuti, Costantino, 2020. "Random time-change with inverses of multivariate subordinators: Governing equations and fractional dynamics," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 6364-6387.
    4. Roman Olson & Soon-Il An & Yanan Fan & Jason P Evans, 2019. "Accounting for skill in trend, variability, and autocorrelation facilitates better multi-model projections: Application to the AMOC and temperature time series," PLOS ONE, Public Library of Science, vol. 14(4), pages 1-24, 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:testjl:v:32:y:2023:i:2:d:10.1007_s11749-022-00844-9. 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.