Tolerance intervals (TIs) are commonly employed in numerous industries, ranging from engineering to pharmaceuticals. However, closed-form TIs are unavailable for most distributions. Although some approximate methods can be used to obtain TIs, coverage probabilities (CPs) of these TIs cannot achieve the nominal level, or can be even far different from the nominal level. In this study, we propose two content-adjusted procedures for TIs based on bootstrap. The first one is based on the bootstrap sample quantile, while the second one is based on the asymptotic normality of empirical distribution. The simulation results show that the two calibration procedures can improve CPs of TIs for some non-normal distributions according to extensive numerical simulations, and they are both proved to be effective through real data examples.

中文翻译：

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通过引导校准进行内容调整的容差间隔

公差区间 (TI) 通常用于从工程到制药的众多行业。但是，大多数发行版都没有封闭式 TI。虽然可以使用一些近似的方法来获得 TI，但这些 TI 的覆盖概率 (CP) 无法达到标称水平，甚至与标称水平相差甚远。在这项研究中，我们提出了两个基于 bootstrap 的 TI 内容调整程序。第一个基于自举样本分位数，而第二个基于经验分布的渐近正态性。仿真结果表明，通过大量的数值模拟，两种标定程序可以提高一些非正态分布的TIs的CPs，并通过实际数据实例证明它们都是有效的。