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

Roxana A. Ion (), Chris A. J. Klaassen () and Edwin R. van den Heuvel ()
Additional contact information
Roxana A. Ion: ASML Veldhoven
Chris A. J. Klaassen: University of Amsterdam
Edwin R. van den Heuvel: Eindhoven University of Technology

TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, 2023, vol. 32, issue 2, No 9, 566-601

Abstract: 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.

Keywords: Cantelli’s inequality; Khintchine representation; Jensen inequality; 60E15; 60E05; 62E99; 62G99; 62P30 (search for similar items in EconPapers)
Date: 2023
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s11749-022-00844-9

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