IDEAS home Printed from https://ideas.repec.org/a/bla/anzsta/v45y2003i2p167-174.html
   My bibliography  Save this article

On the Optimality and Limitations of Buehler Bounds

Author

Listed:
  • Chris J. Lloyd
  • Paul Kabaila

Abstract

In 1957, R.J. Buehler gave a method of constructing honest upper confidence limits for a parameter that are as small as possible subject to a pre‐specified ordering restriction. In reliability theory, these ‘Buehler bounds’ play a central role in setting upper confidence limits for failure probabilities. Despite their stated strong optimality property, Buehler bounds remain virtually unknown to the wider statistical audience. This paper has two purposes. First, it points out that Buehler's construction is not well defined in general. However, a slightly modified version of the Buehler construction is minimal in a slightly weaker, but still compelling, sense. A proof is presented of the optimality of this modified Buehler construction under minimal regularity conditions. Second, the paper demonstrates that Buehler bounds can be expressed as the supremum of Buehler bounds conditional on any nuisance parameters, under very weak assumptions. This result is then used to demonstrate that Buehler bounds reduce to a trivial construction for the location‐scale model. This places important practical limits on the application of Buehler bounds and explains why they are not as well known as they deserve to be.

Suggested Citation

  • Chris J. Lloyd & Paul Kabaila, 2003. "On the Optimality and Limitations of Buehler Bounds," Australian & New Zealand Journal of Statistics, Australian Statistical Publishing Association Inc., vol. 45(2), pages 167-174, June.
  • Handle: RePEc:bla:anzsta:v:45:y:2003:i:2:p:167-174
    DOI: 10.1111/1467-842X.00272
    as

    Download full text from publisher

    File URL: https://doi.org/10.1111/1467-842X.00272
    Download Restriction: no

    File URL: https://libkey.io/10.1111/1467-842X.00272?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. Wang, Weizhen, 2012. "An inductive order construction for the difference of two dependent proportions," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1623-1628.
    2. Kabaila, Paul & Lloyd, Chris J., 2003. "The efficiency of Buehler confidence limits," Statistics & Probability Letters, Elsevier, vol. 65(1), pages 21-28, October.
    3. Lloyd, Chris J., 2008. "More powerful exact tests of binary matched pairs," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2592-2596, November.
    4. Kabaila, Paul, 2008. "Statistical properties of exact confidence intervals from discrete data using studentized test statistics," Statistics & Probability Letters, Elsevier, vol. 78(6), pages 720-727, April.
    5. Lloyd, Chris J., 2005. "Monotonicity of likelihood support bounds for system failure rates," Statistics & Probability Letters, Elsevier, vol. 73(2), pages 91-97, June.

    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:bla:anzsta:v:45:y:2003:i:2:p:167-174. 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: Wiley Content Delivery (email available below). General contact details of provider: http://www.blackwellpublishing.com/journal.asp?ref=1369-1473 .

    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.