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Reciprocal Data Transformations and Their Back-Transforms

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  • Daniel A. Griffith

    (School of Economic, Political, and Policy Sciences, University of Texas at Dallas, Richardson, TX 75080, USA)

Abstract

Variable transformations have a long and celebrated history in statistics, one that was rather academically glamorous at least until generalized linear models theory eclipsed their nurturing normal curve theory role. Still, today it continues to be a covered topic in introductory mathematical statistics courses, offering worthwhile pedagogic insights to students about certain aspects of traditional and contemporary statistical theory and methodology. Since its inception in the 1930s, it has been plagued by a paucity of adequate back-transformation formulae for inverse/reciprocal functions. A literature search exposes that, to date, the inequality E(1/X) ≤ 1/(E(X), which often has a sizeable gap captured by the inequality part of its relationship, is the solitary contender for solving this problem. After documenting that inverse data transformations are anything but a rare occurrence, this paper proposes an innovative, elegant back-transformation solution based upon the Kummer confluent hypergeometric function of the first kind. This paper also derives formal back-transformation formulae for the Manly transformation, something apparently never done before. Much related future research remains to be undertaken; this paper furnishes numerous clues about what some of these endeavors need to be.

Suggested Citation

  • Daniel A. Griffith, 2022. "Reciprocal Data Transformations and Their Back-Transforms," Stats, MDPI, vol. 5(3), pages 1-24, July.
    • Handle: RePEc:gam:jstats:v:5:y:2022:i:3:p:42-737:d:876283
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    References listed on IDEAS

    as
    1. Garcia, Nancy Lopes & Palacios, José Luis, 2001. "On inverse moments of nonnegative random variables," Statistics & Probability Letters, Elsevier, vol. 53(3), pages 235-239, June.
    2. Shuhe Hu & Xinghui Wang & Wenzhi Yang & Xuejun Wang, 2014. "A Note on the Inverse Moment for the Non Negative Random Variables," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 43(8), pages 1750-1757, April.
    3. Lesch, Scott M. & Jeske, Daniel R., 2009. "Some Suggestions for Teaching About Normal Approximations to Poisson and Binomial Distribution Functions," The American Statistician, American Statistical Association, vol. 63(3), pages 274-277.
    4. Daniel A. Griffith, 2013. "Better Articulating Normal Curve Theory for Introductory Mathematical Statistics Students: Power Transformations and Their Back-Transformations," The American Statistician, Taylor & Francis Journals, vol. 67(3), pages 157-169, August.
    5. Freeman, Jade & Modarres, Reza, 2006. "Inverse Box-Cox: The power-normal distribution," Statistics & Probability Letters, Elsevier, vol. 76(8), pages 764-772, April.
    6. Leemis, Lawrence M. & McQueston, Jacquelyn T., 2008. "Univariate Distribution Relationships," The American Statistician, American Statistical Association, vol. 62, pages 45-53, February.
    7. Khuri A. & Casella G., 2002. "The Existence of the First Negative Moment Revisited," The American Statistician, American Statistical Association, vol. 56, pages 44-47, February.
    8. Wenzhi Yang & Shuhe Hu & Xuejun Wang, 2017. "On the asymptotic approximation of inverse moment for non negative random variables," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(16), pages 7787-7797, August.
    9. ChunYuan Wang & Wuyungaowa, 2015. "Recurrence Relation and Accurate Value on Inverse Moment of Discrete Distributions," Journal of Probability and Statistics, Hindawi, vol. 2015, pages 1-7, July.
    10. Wu, Tiee-Jian & Shi, Xiaoping & Miao, Baiqi, 2009. "Asymptotic approximation of inverse moments of nonnegative random variables," Statistics & Probability Letters, Elsevier, vol. 79(11), pages 1366-1371, June.
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