Process capability analysis plays a critical role in quality control by evaluating how well manufacturing processes meet defined specifications. However, traditional process capability indices (PCIs) rely on assumptions of symmetric tolerances and normally distributed data, which often do not hold in real-world applications and can lead to misleading conclusions. To overcome these limitations, we propose two novel classes of PCIs designed specifically for asymmetric tolerances, complemented by parametric estimation procedures and asymptotic confidence limits. To address the issue of non-normal data, we further employ an inverse transformation via constrained B-spline regression, which removes the need for the normality assumption. We demonstrate that our proposed PCIs reduce to traditional indices under symmetric conditions and normal data while extending applicability to a broader range of cases. Numerical simulations and a real-world application in an electronics company confirm the effectiveness and practical utility of our approach.

Multinomial logistic regression models are popular in multicategory classification analysis, but existing models suffer several intrinsic drawbacks. In particular, the parameters cannot be determined uniquely because of the over-specification. Although additional constraints have been imposed to refine the model, such modifications can be inefficient and complicated. In this paper, we propose a novel and efficient simplex-based multinomial logistic regression technique, seamlessly connecting binomial and multinomial cases under a unified framework. Compared with existing models, our model has fewer parameters, is free of any constraints, and can be solved efficiently using the Fisher scoring algorithm. In addition, the proposed model enjoys several theoretical advantages, including Fisher consistency and sharp comparison inequality. Under mild conditions, we establish the asymptotical normality and convergence for the new model, even when the numbers of categories and covariates increase with the sample size. The proposed framework is illustrated by means of extensive simulations and real applications.

The multicategory support vector machine (MSVM) has been widely used for multicategory classification. Despite its widespread popularity, regular MSVM cannot provide direct probabilistic results and suffers from excessive computational cost, as it is formulated on the hinge loss function and it solves a sum-to-zero constrained quadratic programming problem. In this study, we propose a general refinement of regular MSVM, termed as the simplex-based proximal MSVM (SPMSVM). Our SPMSVM uses a novel family of squared error loss functions in place of the hinge loss and it removes the explicit sum-to-zero constraint by the simplex structure. Consequently, the SPMSVM only requires solving an unconstrained linear system, leading to closed-form solutions. In addition, the SPMSVM can be cast into a weighted regression problem so that it is scalable for large-scale applications. Moreover, the SPMSVM naturally yields an estimate of the conditional category probability, which is more informative than regular MSVM. Theoretically, the SPMSVM is shown to include many existing MSVMs as its special cases, and its asymptotic and finite-sample statistical properties are well established. Simulations and real examples show that the proposed SPMSVM is a stable, scalable and competitive classifier.