On the Optimality and Limitations of Buehler Bounds
On the Optimality and Limitations of Buehler Bounds
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DOI：
10.1111/1467-842x.00272
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发表时间：
2003-06
影响因子：
1.1
通讯作者：
C. Lloyd;Paul Kabaila
中科院分区：
文献类型：
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作者：
C. Lloyd;Paul Kabaila
1957年，R. J. Buehler给出了一种方法，可以为一个参数构造诚实的置信上限，该参数在预先指定的排序限制下尽可能小。在可靠性理论中，这些“Buehler界限”在设置失效概率的置信上限方面起着核心作用。尽管他们声明的强最优属性，Buehler界限仍然几乎不为更广泛的统计受众所知。本文有两个目的。首先，指出Buehler的构造在一般意义上并没有得到很好的定义。然而，一个稍微修改版本的Buehler建设是最小的，在一个稍微弱，但仍然引人注目的意义。证明了这种修改的Buehler建设最小正则性条件下的最优性。其次，证明了在非常弱的假设下，Buehler界可以表示为Buehler界的上确界条件的任何滋扰参数。这个结果然后被用来证明Buehler边界减少到一个平凡的位置-尺度模型的建设。这对Buehler界限的应用提出了重要的实际限制，并解释了为什么它们不像它们应该的那样广为人知。
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.