IDEAS home Printed from https://ideas.repec.org/a/jss/jstsof/v016i04.html
   My bibliography  Save this article

Ratios of Normal Variables

Author

Listed:
  • Marsaglia, George

Abstract

This article extends and amplifies on results from a paper of over forty years ago. It provides software for evaluating the density and distribution functions of the ratio z/w for any two jointly normal variates z,w, and provides details on methods for transforming a general ratio z/w into a standard form, (a+x)/(b+y) , with x and y independent standard normal and a, b non-negative constants. It discusses handling general ratios when, in theory, none of the moments exist yet practical considerations suggest there should be approximations whose adequacy can be verified by means of the included software. These approximations show that many of the ratios of normal variates encountered in practice can themselves be taken as normally distributed. A practical rule is developed: If a

Suggested Citation

  • Marsaglia, George, 2006. "Ratios of Normal Variables," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 16(i04).
    • Handle: RePEc:jss:jstsof:v:016:i04
    DOI: http://hdl.handle.net/10.18637/jss.v016.i04
    as

    Download full text from publisher

    File URL: https://www.jstatsoft.org/index.php/jss/article/view/v016i04/v16i04.pdf
    Download Restriction: no
    File URL: https://www.jstatsoft.org/index.php/jss/article/downloadSuppFile/v016i04/zoverw.c.zip
    Download Restriction: no
    File URL: https://libkey.io/http://hdl.handle.net/10.18637/jss.v016.i04?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Tom Burr & Elisa Bonner & Kamil Krzysztoszek & Claude Norman, 2019. "Setting Alarm Thresholds in Measurements with Systematic and Random Errors," Stats, MDPI, vol. 2(2), pages 1-13, May.
    2. Carlotta Galeone & Angiola Pollastri, 2012. "Confidence intervals for the ratio of two means using the distribution of the quotient of two normals," Statistics in Transition new series, Główny Urząd Statystyczny (Polska), vol. 13(3), pages 451-472, December.
    3. Dennis Wichelns, 2015. "Water productivity and water footprints are not helpful in determining optimal water allocations or efficient management strategies," Water International, Taylor & Francis Journals, vol. 40(7), pages 1059-1070, November.
    4. Hsin-Neng Hsieh & Hung-Yi Lu, 2020. "The generalized inference on the ratio of mean differences for fraction retention noninferiority hypothesis," PLOS ONE, Public Library of Science, vol. 15(6), pages 1-12, June.
    5. Clark, Adam Thomas & Neuhauser, Claudia, 2018. "Harnessing uncertainty to approximate mechanistic models of interspecific interactions," Theoretical Population Biology, Elsevier, vol. 123(C), pages 35-44.
    6. Stanley Luck, 2022. "A parametric framework for multidimensional linear measurement error regression," PLOS ONE, Public Library of Science, vol. 17(1), pages 1-22, January.
    7. Eloísa Díaz-Francés & Francisco Rubio, 2013. "On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables," Statistical Papers, Springer, vol. 54(2), pages 309-323, May.
    8. Caginalp, Carey & Caginalp, Gunduz, 2018. "The quotient of normal random variables and application to asset price fat tails," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 499(C), pages 457-471.
    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. Guy P. Nason & Ben Powell & Duncan Elliott & Paul A. Smith, 2017. "Should we sample a time series more frequently?: decision support via multirate spectrum estimation," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 180(2), pages 353-407, February.
    11. Caginalp, Carey & Caginalp, Gunduz, 2019. "Price equations with symmetric supply/demand; implications for fat tails," Economics Letters, Elsevier, vol. 176(C), pages 79-82.
    12. Michael Keane & Timothy Neal, 2021. "A New Perspective on Weak Instruments," Discussion Papers 2021-05a, School of Economics, The University of New South Wales.
    13. Michael Keane & Timothy Neal, 2021. "A Practical Guide to Weak Instruments," Discussion Papers 2021-05b, School of Economics, The University of New South Wales.
    14. Bagos Pantelis G, 2008. "A Unification of Multivariate Methods for Meta-Analysis of Genetic Association Studies," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 7(1), pages 1-35, October.
    15. Erhard Reschenhofer, 2017. "Using Ratios of Successive Returns for the Estimation of Serial Correlation in Return Series," Noble International Journal of Economics and Financial Research, Noble Academic Publsiher, vol. 2(9), pages 125-130, September.
    16. Frantisek Duris & Juraj Gazdarica & Iveta Gazdaricova & Lucia Strieskova & Jaroslav Budis & Jan Turna & Tomas Szemes, 2018. "Mean and variance of ratios of proportions from categories of a multinomial distribution," Journal of Statistical Distributions and Applications, Springer, vol. 5(1), pages 1-20, December.
    17. Gatta, Valerio & Marcucci, Edoardo & Scaccia, Luisa, 2015. "On finite sample performance of confidence intervals methods for willingness to pay measures," Transportation Research Part A: Policy and Practice, Elsevier, vol. 82(C), pages 169-192.
    18. Stokes, Barrie, 2012. "mathStatica 2.5," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 47(s01).
    19. Alvarez, Eduardo J. & Ribaric, Adrijan P., 2018. "An improved-accuracy method for fatigue load analysis of wind turbine gearbox based on SCADA," Renewable Energy, Elsevier, vol. 115(C), pages 391-399.
    20. Gunduz Caginalp, 2020. "Fat tails arise endogenously in asset prices from supply/demand, with or without jump processes," Papers 2011.08275, arXiv.org, revised Mar 2021.

    More about this item

    Statistics

    Access and download statistics

    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:jss:jstsof:v:016:i04. 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.

    We have no bibliographic references for this item. You can help adding them by using 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: Christopher F. Baum (email available below). General contact details of provider: http://www.jstatsoft.org/ .

    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.