Measurement System R&R Analysis for Zero-Inflated Correlated Defect Counts in Semiconductor Wafer AOI Inspection

This study proposes a novel measurement system repeatability and reproducibility (R&R) framework for zero-inflated correlated defect-count data in semiconductor wafer automated optical inspection (AOI). In advanced semiconductor manufacturing environments, AOI systems are extensively used to detect wafer defects such as particles, scratches, and structural abnormalities. However, conventional Gauge R&R methods are primarily developed for continuous Gaussian-type measurements and are therefore not fully appropriate for high-yield semiconductor inspection data characterized by discrete defect counts, excessive zero observations, and correlated defect categories. To address these limitations, this study develops a zero-inflated bivariate Poisson (ZIBP) measurement system model capable of simultaneously capturing correlated defect-generation mechanisms and structural zero-defect states. A latent-variable representation is introduced to model shared and category-specific defect sources, while a zero-inflation mechanism accounts for defect-free wafer observations commonly encountered in precision manufacturing. An expectation-maximization (EM) algorithm is further developed for parameter estimation, including latent common defect counts and structural-zero probabilities. Based on the fitted model, repeatability variance, reproducibility variance, total measurement variation, and Percent R&R are estimated under the proposed probabilistic framework. In addition, bootstrap resampling is employed to construct confidence intervals for the proposed R&R measures. Theoretical properties of the proposed framework, including covariance structure, identifiability, EM monotonicity, estimator consistency, and asymptotic behavior of the Percent R&R estimator, are analytically established. The proposed framework extends traditional Gauge R&R analysis from continuous Gaussian measurements to zero-inflated correlated count-type defect inspection data and provides a statistically rigorous methodology for evaluating AOI measurement system reliability in semiconductor wafer manufacturing environments.