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On the Optimality and Limitations of Buehler Bounds

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  • 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
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    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. Lloyd, Chris J., 2008. "More powerful exact tests of binary matched pairs," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2592-2596, November.
    3. 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.
    4. Kabaila, Paul & Lloyd, Chris J., 2003. "The efficiency of Buehler confidence limits," Statistics & Probability Letters, Elsevier, vol. 65(1), pages 21-28, October.
    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.

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