國立政治大學統計學系

學  術  演  講

主講人：楊宏基 助理教授
國立東華大學應用數學系

題      目：Density–Quantile Tail-Adaptive Control Limits with EPC Calibration for Limited Phase-I Samples

時      間：民國115年10月12日 (星期一) 下午1：30 

地      點：國立政治大學逸仙樓050101教室

摘      要：

Designing control limits for a very low false-alarm probability is challenging when the process distribution is unknown and only a limited Phase-I reference sample is available: the required quantiles may lie beyond the smallest and largest observations. This talk presents a semiparametric method for estimating these limits while accounting for reference-sample uncertainty. Log density–quantile (LDQ) regression estimates separate Parzen exponents that characterize the lower and upper tails using observations in boundary neighborhoods. These estimates guide an adaptive fractional-order-statistics (FOS) estimator that interpolates within the observed range and extrapolates beyond the sample extremes. The exceedance probability criterion (EPC) specifies a target probability, over repeated Phase-I samples, that the resulting in-control false-alarm probability does not exceed a prescribed tolerance. Beta-based EPC calibration selects the quantile levels, while an empirically derived rule adjusts the confidence target to the available sample size. The same rule is evaluated across five benchmark distributions. Under stated regularity conditions, we establish consistency of the tail-exponent estimators and first-order accuracy of tail extrapolation. Nonparametric bootstrap intervals quantify uncertainty in the estimated exponents and control limits. Simulations with sample sizes from 50 to 2500 and a false-alarm tolerance of 0.0027 examine normal, chi-squared, beta, lognormal, and Student’s t distributions. At n = 2500, estimated EPC attainment ranges from 0.964 to 0.982 for a target of 0.95. Under the normal benchmark, attainment is 0.967, compared with 0.916 and 0.937 for extreme-value methods based on the Pickands and moment estimators, respectively. An application to four SECOM semiconductor process variables illustrates tail-specific control limits and their estimation uncertainty. The method is implemented in the R package adaFOS.

Keywords: Density–quantile function; tail exponent; exceedance probability criterion; bootstrap confidence interval; statistical process monitoring