淘汰蒙特卡洛：均值-散布放行规则的精确接受概率

中文摘要 AI 辅助

一大类受监管的放行决策，当且仅当 $n$ 个单位的（样本均值，样本标准差）这一对值落在某个固定平面区域内时，才接受该生产批次。含量均匀度（USP <905>）、能力放行、变量抽样以及限内百分比高速公路验收均采用此形式。据我们所知，在当前实践中，假设的单位级分布下的接受概率并未被精确评估：对于一般的总体，$(\bar X,s)$ 的联合有限 $n$ 分布是一个受约束的 $n$ 重积分（Craig 1932 年针对 $n=3,4$；Springer 1953 年针对一般 $n$），该积分不存在初等化简。然而，该概率由加性对 $(T_1,T_2)=(\sum_{i=1}^{n} X_i,\sum_{i=1}^{n} X_i^2)$ 的分布决定，其特征函数是单观测变换的 $n$ 次幂。因此，一次二维傅里叶反演即可为任意光滑总体精确地（作为恒等式）并在数值上（在维度始终为二的网格上，无论 $n$ 取何值）给出该概率——这是对模拟和正态性假设的一种确定性替代方案。该恒等式是精确的；我们所评估的是其有限网格反演，其误差我们按示例报告而非给出界限。当 $n=3$ 且总体为均匀分布时，$(\bar X,s)$ 的联合分布、接受概率和能力分布可得到闭式解；我们推导出这些结果，将其用作验证阶梯的锚点，然后在非正态总体下以其规定的样本量（$n=10$-$30$）端到端地处理三个规则。与精确答案相比，基于正态理论的现有方法所犯的误差足以改变计划所针对的风险，并且在工作曲线上两个方向都会出错，因此没有任何单一常数可以纠正它。一般而言，只有完整的分布才足够。

英文摘要

A large family of regulated release decisions accepts a production lot if and only if the pair (sample mean, sample standard deviation) of $n$ units falls in a fixed plane region. Content uniformity (USP <905>), capability release, variables sampling and percent-within-limits highway acceptance all take this form. The acceptance probability under a hypothesized unit-level law is not, so far as we are aware, evaluated exactly in current practice: for a general parent the joint finite-$n$ law of $(\bar X,s)$ is a constrained $n$-fold integral (Craig 1932 at $n=3,4$; Springer 1953 at general $n$) which admits no elementary reduction. Yet that probability is fixed by the law of the additive pair $(T_1,T_2)=(\sum_{i=1}^{n} X_i,\sum_{i=1}^{n} X_i^2)$, whose characteristic function is the $n$-th power of a one-observation transform. One two-dimensional Fourier inversion therefore delivers it for an arbitrary smooth parent - exactly, as an identity, and numerically on a grid whose dimension stays two whatever $n$ is - a deterministic alternative to both the simulation and the Normality assumption. The identity is exact; what we evaluate is a finite-grid inversion of it, whose error we report per example rather than bound. At $n=3$ the Uniform parent yields the joint law of $(\bar X,s)$, the acceptance probability and the capability law in closed form; we derive these, use them as the anchor of a validation ladder, and then work three rules end to end under non-normal parents at their prescribed sample sizes ($n=10$-$30$). Against the exact answer the Normal-theory incumbent errs by enough to change the risk a plan is designed against, and errs in both directions along the operating curve, so no single constant corrects it. In general only the full distribution suffices.

发表机构

• Gilead Sciences（吉利德科学）

机构由 AI 辅助整理，请以论文原文为准。

补充信息

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63 pages, 10 figures