Extreme value theory for individuals control charts: a semiparametric approach to ensuring in-control performance

This study introduces a semiparametric method for designing individuals control charts in Phase I statistical process monitoring, aimed at ensuring guaranteed in-control performance. Motivated by the limitations of parametric and nonparametric methods under small to moderate sample sizes, the proposed approach combines extreme value theory with the exceedance probability criterion to construct control limits that adjust for estimation variability. The methodology uses Pickands and moment estimators to model extreme tail behavior, offering a more reliable solution than purely data-driven nonparametric methods. Simulation results show that a sample size of approximately $n=1112$ is sufficient to achieve a nominal coverage probability of $P_n=0.90$ . The method demonstrates strong and consistent performance across a variety of distributions, avoids the need for computationally intensive bootstrap procedures, and provides an R package for implementation to support reproducibility and real-world applications, such as semiconductor manufacturing.