A reasonable confidence interval should have a confidence coefficient no less than the given nominal level and a small expected length to reliably and accurately estimate the parameter of interest, and the bootstrap interval is considered to be an efficient interval estimation technique. In this paper, we offer a first attempt at computing the coverage probability and expected length of a parametric or percentile bootstrap interval by exact probabilistic calculation for any fixed sample size. This method is applied to the basic bootstrap intervals for functions of binomial proportions and a normal mean. None of these intervals, however, are found to have a correct confidence coefficient, which leads to illogical conclusions including that the bootstrap interval is narrower than the z-interval when estimating a normal mean. This raises a general question of how to utilize bootstrap intervals appropriately in practice since the sample size is typically fixed.

翻译：合理的置信区间应具备不低于给定名义水平的置信系数和较小的期望长度，以可靠准确地估计目标参数，而Bootstrap区间被认为是一种高效的区间估计技术。本文首次通过精确概率计算，针对任意固定样本量，给出了参数型或百分位型Bootstrap区间的覆盖概率与期望长度的计算方法。该方法被应用于二项比例函数和正态均值的Bootstrap基本区间。然而，这些区间均未发现具有正确的置信系数，导致在估计正态均值时出现Bootstrap区间窄于z区间的逻辑悖论。由于实际应用中样本量往往是固定的，这引发了一个普遍性问题：如何恰当地在实践使用Bootstrap区间。