Nonlinear Amplification of Finite-Sample Uncertainty in Capability-Based Decisions

Abstract: 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.