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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
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    References listed on IDEAS

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    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. 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.
    4. 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.
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