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A general treatment of alternative expectation formulae

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  • Liu, Yang

Abstract

The moment of a positive random variable can be obtained by using the survival function. In this paper, we extend existing studies by giving formulae of the expectation of a more general function with respect to a random variable by using either cumulative distribution function or survival function. Both univariate and multivariate examples are given. These formulae can evaluate the expectation when obtaining the probability density/mass function is difficult.

Suggested Citation

  • Liu, Yang, 2020. "A general treatment of alternative expectation formulae," Statistics & Probability Letters, Elsevier, vol. 166(C).
    • Handle: RePEc:eee:stapro:v:166:y:2020:i:c:s0167715220301668
    DOI: 10.1016/j.spl.2020.108863
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    References listed on IDEAS

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    1. Lo, Ambrose, 2017. "Functional generalizations of Hoeffding’s covariance lemma and a formula for Kendall’s tau," Statistics & Probability Letters, Elsevier, vol. 122(C), pages 218-226.
    2. Ambrose Lo, 2019. "Demystifying the Integrated Tail Probability Expectation Formula," The American Statistician, Taylor & Francis Journals, vol. 73(4), pages 367-374, October.
    3. Liang Hong, 2015. "Another Remark on the Alternative Expectation Formula," The American Statistician, Taylor & Francis Journals, vol. 69(3), pages 157-159, August.
    4. Pingfan Song & Shaochen Wang, 2019. "On the moment generating functions for integer valued random variables," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 48(20), pages 5169-5174, October.
    5. Song, Pingfan & Tan, Changchun & Wang, Shaochen, 2019. "On the moment generating function for random vectors via inverse survival function," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 345-350.
    6. Haruhiko Ogasawara, 2020. "Alternative expectation formulas for real-valued random vectors," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 49(2), pages 454-470, January.
    7. M.C. Jones, 2019. "Letter to the Editor," The American Statistician, Taylor & Francis Journals, vol. 73(1), pages 105-105, January.
    8. Hok Shing Kwong & Saralees Nadarajah, 2018. "Expectation formulas for integer valued multivariate random variables," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 47(22), pages 5514-5518, November.
    9. Subhabrata Chakraborti & Felipe Jardim & Eugenio Epprecht, 2019. "Higher-Order Moments Using the Survival Function: The Alternative Expectation Formula," The American Statistician, Taylor & Francis Journals, vol. 73(2), pages 191-194, April.
    10. Liang Hong, 2012. "A Remark on the Alternative Expectation Formula," The American Statistician, Taylor & Francis Journals, vol. 66(4), pages 232-233, November.
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    Cited by:

    1. Edelmann, Dominic & Richards, Donald & Royen, Thomas, 2023. "Product inequalities for multivariate Gaussian, gamma, and positively upper orthant dependent distributions," Statistics & Probability Letters, Elsevier, vol. 197(C).
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