Theory of Probability and Mathematical Statistics
A Markovian Gauss inequality for asymmetric deviations from the mode of symmetric unimodal distributions
Chris A.J. Klaassen
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Abstract: For a random variable with a unimodal distribution and finite second moment Gauß (1823) proved a sharp bound on the probability of the random variable to be outside a symmetric interval around its mode. An alternative proof for it under the assumption of a finite r-th absolute moment is given (r≥1), based on the Khintchine representation of unimodal random variables. A special instance of the resulting Narumi–Gauß inequality is the one with finite first absolute moment, which might be called a Markovian Gauß inequality.
For symmetric unimodal distributions with finite second moment Semenikhin (2019) generalized the Gauß inequality to arbitrary intervals. For the class of symmetric unimodal distributions with finite first absolute moment we construct a Markovian version of it. Related inequalities of Volkov (1969) and Sellke and Sellke (1997) will be discussed as well.
Keywords: Markovian Gauß inequality, Khintchine representation, Volkov inequality
Bibliography: I. J. Bienaymé, Considérations à l’appui de la decouverte de Laplace sur la loi de probabilité dans la méthode des moindres carrés, C.R. Acad. Sci. Paris 37 (1853), 5–13. Reprinted: Liouville’s J. Math. Pures Appl. (2) 12 (1867), 158–176.
P. L. Chebyshev, Des valeurs moyennes, Liouville’s J. Math. Pures Appl. (2) 12, (1867), 177–184.
Harald Cramér, Mathematical Methods of Statistics, Princeton Mathematical Series, vol. 9, Princeton University Press, Princeton, NJ, 1946. MR 16588
C. F. Gauß, Theoria combinationis observationum erroribus minimis obnoxiae, pars prior, Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores (1823), 1–58.
C. C. Heyde and E. Seneta, Studies in the history of probability and statistics. XXXI. The simple branching process, a turning point test and a fundamental inequality: a historical note on I. J. Bienaymé, Biometrika 59 (1972), 680–683. MR 341541, DOI 10.1093/biomet/59.3.680
Gerard Hooghiemstra and Piet Van Mieghem, An inequality of Gauss, Nieuw Arch. Wiskd. (5) 16 (2015), no. 2, 123–126. MR 3381944
Roxana A. Ion, Chris A. J. Klaassen, and Edwin R. van den Heuvel, Sharp inequalities of Bienaymé-Chebyshev and Gaußtype for possibly asymmetric intervals around the mean, TEST 32 (2023), no. 2, 566–601. MR 4621162, DOI 10.1007/s11749-022-00844-9
Samuel Karlin and William J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. MR 204922
A. Ya. Khintchine, On unimodal distributions, Izv. Nauchno-Issled. Inst. Mat. Mech. Tomsk. Gos. Univ. 2 (1938), 1–7.
S. Narumi, On Further Inequalities with Possible Application to Problems in the Theory of Probability, Biometrika 15 (1923), 245–253.
Friedrich Pukelsheim, The three sigma rule, Amer. Statist. 48 (1994), no. 2, 88–91. MR 1292524, DOI 10.2307/2684253
I. Richard Savage, Probability inequalities of the Tchebycheff type, J. Res. Nat. Bur. Standards Sect. B 65B (1961), 211–222. MR 125603, DOI 10.6028/jres.065B.020
Thomas M. Sellke and Sarah H. Sellke, Chebyshev inequalities for unimodal distributions, Amer. Statist. 51 (1997), no. 1, 34–40. MR 1440938, DOI 10.2307/2684690
K. V. Semenikhin, A two-sided probability bound for a symmetric unimodal random variable, Avtomat. i Telemekh. 3 (2019), 103–122 (Russian, with Russian summary); English transl., Autom. Remote Control 80 (2019), no. 3, 474–489. MR 3935472
Carl Friedrich Gauss, Theoria combinationis observationum erroribus minimis obnoxiae/Theory of the combination of observations least subject to errors, Classics in Applied Mathematics, vol. 11, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. Pars prior. Pars posterior. Supplementum/Part One. Part Two. Supplement; Dual Latin-English text; Translated and with an introduction and afterword by G. W. Stewart. MR 1329543, DOI 10.1137/1.9781611971248
D. F. Vysočanskiĭ and Ju. Ī. Petunīn, Proof of the 3σ rule for unimodal distributions, Teor. Veroyatnost. i Mat. Statist. 21 (1979), 23–35, 163 (Russian, with English summary). MR 550239
V. N. Volkov, Inequalities connected with an inequality of Gauss, Volž. Mat. Sb. 7 (1969), 11–13. (loose errata) (Russian). MR 280663
