Well-recommended methods of forming “confidence intervals” for a binomial proportion give interval estimates that do not actually meet the definition of a confidence interval, in that their coverages are sometimes lower than the nominal confidence level. The methods are favoured because their intervals have a shorter average length than the Clopper–Pearson (gold-standard) method, whose intervals really are confidence intervals. As the definition of a confidence interval is not being adhered to, another criterion for forming interval estimates for a binomial proportion is needed. In this paper, we suggest a new criterion for forming one-sided intervals and equal-tail two-sided intervals. Methods which meet the criterion are said to yield locally correct confidence intervals. We propose a method that yields such intervals and prove that its intervals have a shorter average length than those of any other method that meets the criterion. Compared with the Clopper–Pearson method, the proposed method gives intervals with an appreciably smaller average length. For confidence levels of practical interest, the mid- method also satisfies the new criterion and has its own optimality property. It gives locally correct confidence intervals that are only slightly wider than those of the new method.

中文翻译：

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二项式比例的局部正确置信区间：区间估计器的新标准

推荐的形成二项式比例“置信区间”的方法给出的区间估计实际上并不符合置信区间的定义，因为它们的覆盖范围有时低于名义置信水平。这些方法之所以受到青睐，是因为它们的区间平均长度比克洛珀-皮尔逊（黄金标准）方法更短，后者的区间实际上是置信区间。由于不遵守置信区间的定义，因此需要另一个用于形成二项式比例的区间估计的标准。在本文中，我们提出了一种形成单边区间和等尾两侧区间的新标准。满足该标准的方法据说可以产生局部正确的置信区间。我们提出了一种产生此类区间的方法，并证明其区间的平均长度比任何其他满足标准的方法的平均长度更短。与 Clopper-Pearson 方法相比，该方法给出的区间平均长度明显更小。对于实际利益的置信水平，中期该方法也满足新的准则并具有其自身的最优性。它给出了局部正确的置信区间，仅比新方法的置信区间稍宽。