Reciprocal Data Transformations and Their Back-Transforms
School of Economic, Political, and Policy Sciences, University of Texas at Dallas, Richardson, TX 75080, USA
Academic Editor: Wei Zhu
Stats 2022, 5(3), 714-737; https://doi.org/10.3390/stats5030042
Received: 7 July 2022
/
Revised: 26 July 2022
/
Accepted: 28 July 2022
/
Published: 30 July 2022
(This article belongs to the Section Statistical Methods)
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.
View Full-Text
Keywords:
back-transformation; Box–Cox transformation; inverse random variables; manly transformation; power transformation; reciprocal random variables
▼
Show Figures
Figure 1
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
MDPI and ACS Style
Griffith, D.A. Reciprocal Data Transformations and Their Back-Transforms. Stats 2022, 5, 714-737. https://doi.org/10.3390/stats5030042
AMA Style
Griffith DA. Reciprocal Data Transformations and Their Back-Transforms. Stats. 2022; 5(3):714-737. https://doi.org/10.3390/stats5030042
Chicago/Turabian StyleGriffith, Daniel A. 2022. "Reciprocal Data Transformations and Their Back-Transforms" Stats 5, no. 3: 714-737. https://doi.org/10.3390/stats5030042
Find Other Styles
