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Some Improvements on Markov's Theorem with Extensions

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  • Haruhiko Ogasawara

Abstract

Markov's theorem for an upper bound of the probability related to a nonnegative random variable has been improved using additional information in almost the nontrivial entire range of the variable. In the improvement, Cantelli's inequality is applied to the square root of the original variable, whose expectation is finite when that of the original variable is finite. The improvement has been extended to lower bounds and monotonic transformations of the original variable. The improvements are used in Chebyshev's inequality and its multivariate version.

Suggested Citation

  • Haruhiko Ogasawara, 2020. "Some Improvements on Markov's Theorem with Extensions," The American Statistician, Taylor & Francis Journals, vol. 74(3), pages 218-225, July.
    • Handle: RePEc:taf:amstat:v:74:y:2020:i:3:p:218-225
    DOI: 10.1080/00031305.2018.1497539
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    Cited by:

    1. Budny, Katarzyna, 2022. "The variance of the power of a shifted random variable with applications," Statistics & Probability Letters, Elsevier, vol. 182(C).
    2. Budny, Katarzyna, 2022. "Improved probability inequalities for Mardia’s coefficient of kurtosis," Statistics & Probability Letters, Elsevier, vol. 191(C).

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