IDEAS home Printed from https://ideas.repec.org/a/spr/metrik/v82y2019i8d10.1007_s00184-019-00710-z.html
   My bibliography  Save this article

Cumulative distribution functions for the five simplest natural exponential families

Author

Listed:
  • Gérard Letac

    (Université Paul Sabatier)

Abstract

Consider an exponential family F on the set of non-negative integers indexed by the parameter a. The cumulative distribution function of an element of F estimated on k is both a function of a and k. Assume that the derivative of this function with respect to a is the product of three things: a function of k, a function of a and the function a to the power k. We show that this assumption implies that the exponential family is either a binomial, or the Poisson, or a negative binomial family. Next, we study an analogous property for continuous distributions and we find that it is satisfied if and only the families are either Gaussian or Gamma. Ultimately, the proofs rely on the fact that only Möbius functions preserve the cross ratio.

Suggested Citation

  • Gérard Letac, 2019. "Cumulative distribution functions for the five simplest natural exponential families," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(8), pages 891-902, November.
    • Handle: RePEc:spr:metrik:v:82:y:2019:i:8:d:10.1007_s00184-019-00710-z
    DOI: 10.1007/s00184-019-00710-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00184-019-00710-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00184-019-00710-z?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    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:spr:metrik:v:82:y:2019:i:8:d:10.1007_s00184-019-00710-z. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.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.