This paper studies the propagation of finite-sample uncertainty under nonlinear transformations commonly used in statistical decision systems. In particular, we consider process capability indices, which are widely used in manufacturing practice but are estimated from finite samples, rendering the resulting approval decisions inherently uncertain. We show that such uncertainty cannot be fully explained by estimator variability alone, but is substantially influenced by a nonlinear amplification mechanism through which capability uncertainty is transformed into defect-risk metrics. While capability estimators vary approximately linearly with process dispersion, defect probabilities depend on tail curvature, causing small estimation errors to be disproportionately amplified in measures such as defect probability and parts-per-million (PPM) rates. Consequently, capability assessments that appear stable in index space may exhibit substantial variability in defect-risk space, particularly near decision thresholds. This insight provides a unified explanation of finite-sample decision instability, motivates reliability-aware decision formulations, and links sample-size requirements directly to decision reliability. Monte Carlo simulations and industrial data analyses validate the proposed mechanism and demonstrate its practical implications, including the impact of distributional assumptions on defect-risk estimation.

翻译：本文研究了统计决策系统中常用非线性变换下有限样本不确定性的传播机制。特别地，我们聚焦于过程能力指数——该指标在制造业实践中广泛应用，但由于基于有限样本估计，导致由此产生的审批决策天然具有不确定性。研究表明，此类不确定性无法仅用估计量变异充分解释，而是受非线性放大机制显著影响——该机制将能力不确定性转化为缺陷风险指标。尽管能力估计量随过程离散程度近似线性变化，但缺陷概率取决于尾部分布曲率，使得微小估计误差在缺陷概率和百万分之缺陷数(PPM)等指标中被非线性放大。因此，在指数空间看似稳定的能力评估，在缺陷风险空间可能呈现显著波动，尤其在决策阈值附近。这一发现为有限样本决策不稳定性提供了统一解释，催生了可靠性感知的决策框架，并将样本量要求与决策可靠性直接关联。蒙特卡洛模拟与工业数据分析验证了所提机制的有效性，并揭示了其实际应用意义，包括分布假设对缺陷风险估计的影响。