IDEAS home Printed from https://ideas.repec.org/a/gam/jstats/v5y2022i3p42-737d876283.html
   My bibliography  Save this article

Reciprocal Data Transformations and Their Back-Transforms

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2571-905X/5/3/42/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2571-905X/5/3/42/
    Download Restriction: no
    ---><---

    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. Freeman, Jade & Modarres, Reza, 2006. "Inverse Box-Cox: The power-normal distribution," Statistics & Probability Letters, Elsevier, vol. 76(8), pages 764-772, April.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. Leemis, Lawrence M. & McQueston, Jacquelyn T., 2008. "Univariate Distribution Relationships," The American Statistician, American Statistical Association, vol. 62, pages 45-53, February.
    10. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Hongyan Fang & Saisai Ding & Xiaoqin Li & Wenzhi Yang, 2020. "Asymptotic Approximations of Ratio Moments Based on Dependent Sequences," Mathematics, MDPI, vol. 8(3), pages 1-18, March.
    2. Daniel A. Griffith, 2014. "Reply," The American Statistician, Taylor & Francis Journals, vol. 68(1), pages 67-69, February.
    3. Wang, Xuejun & Hu, Shuhe & Yang, Wenzhi & Ling, Nengxiang, 2010. "Exponential inequalities and inverse moment for NOD sequence," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 452-461, March.
    4. Shi, Xiaoping & Wu, Yuehua & Liu, Yu, 2010. "A note on asymptotic approximations of inverse moments of nonnegative random variables," Statistics & Probability Letters, Elsevier, vol. 80(15-16), pages 1260-1264, August.
    5. Klein, Ingo, 2012. "Quasi-arithmetische Mittelwerte und Normalverteilung," Discussion Papers 89/2010, Friedrich-Alexander University Erlangen-Nuremberg, Chair of Statistics and Econometrics.
    6. Vladimir Hlasny, 2021. "Parametric representation of the top of income distributions: Options, historical evidence, and model selection," Journal of Economic Surveys, Wiley Blackwell, vol. 35(4), pages 1217-1256, September.
    7. Escanciano, Juan Carlos, 2023. "Irregular identification of structural models with nonparametric unobserved heterogeneity," Journal of Econometrics, Elsevier, vol. 234(1), pages 106-127.
    8. Marcelo Fernandes & Eduardo Mendes & Olivier Scaillet, 2015. "Testing for symmetry and conditional symmetry using asymmetric kernels," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(4), pages 649-671, August.
    9. Carson, Richard T. & Czajkowski, Mikołaj, 2019. "A new baseline model for estimating willingness to pay from discrete choice models," Journal of Environmental Economics and Management, Elsevier, vol. 95(C), pages 57-61.
    10. Greenberg, Spencer & Mohri, Mehryar, 2014. "Tight lower bound on the probability of a binomial exceeding its expectation," Statistics & Probability Letters, Elsevier, vol. 86(C), pages 91-98.
    11. Seyyed Ali Zeytoon Nejad Moosavian, 2022. "The Visual Decoding of the Wheel of Duality in Consumer Theory in Modern Microeconomics," Papers 2209.02839, arXiv.org.
    12. Janette Larney & Gerrit Lodewicus Grobler & James Samuel Allison, 2022. "Introducing Two Parsimonious Standard Power Mixture Models for Bimodal Proportional Data with Application to Loss Given Default," Mathematics, MDPI, vol. 10(23), pages 1-19, November.
    13. Songyot Kitthamkesorn & Anthony Chen, 2024. "Stochastic User Equilibrium Model with a Bounded Perceived Travel Time," Papers 2402.18435, arXiv.org.
    14. Planas, Christophe & Rossi, Alessandro, 2018. "The slice sampler and centrally symmetric distributions," JRC Working Papers in Economics and Finance 2018-11, Joint Research Centre, European Commission.
    15. Kleijnen, J.P.C., 2007. "Simulation Experiments in Practice : Statistical Design and Regression Analysis," Discussion Paper 2007-30, Tilburg University, Center for Economic Research.
    16. Eurek, Kelly & Sullivan, Patrick & Gleason, Michael & Hettinger, Dylan & Heimiller, Donna & Lopez, Anthony, 2017. "An improved global wind resource estimate for integrated assessment models," Energy Economics, Elsevier, vol. 64(C), pages 552-567.
    17. Kleijnen, J.P.C., 2006. "White Noise Assumptions Revisited : Regression Models and Statistical Designs for Simulation Practice," Other publications TiSEM d8c37ad3-f9a5-4824-986d-2, Tilburg University, School of Economics and Management.
    18. L. Raso & S. V. Weijs & M. Werner, 2018. "Balancing Costs and Benefits in Selecting New Information: Efficient Monitoring Using Deterministic Hydro-economic Models," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 32(1), pages 339-357, January.
    19. Cribari-Neto, Francisco & Garcia, Nancy Lopes & Vasconcellos, Klaus L. P., 2000. "A Note on Inverse Moments of Binomial Variates," Brazilian Review of Econometrics, Sociedade Brasileira de Econometria - SBE, vol. 20(2), November.
    20. Proietti, Tommaso & Lütkepohl, Helmut, 2013. "Does the Box–Cox transformation help in forecasting macroeconomic time series?," International Journal of Forecasting, Elsevier, vol. 29(1), pages 88-99.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jstats:v:5:y:2022:i:3:p:42-737:d:876283. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.