Process capability indices (PCIs) have been widely used in industries for assessing the capability of manufacturing processes. Castagliola and Castellanos (Quality Technology and Quantitative Management 2005, 2(2):201–220), viewing that there were no clear links between the definition of the existing multivariate PCIs and theoretical proportion of nonconforming product items, defined a bivariate Cpk and Cp (denoted by BCpk and BCp, respectively) based on the proportions of nonconforming product items over four convex polygons for bivariate normal processes with a rectangular specification region. In this paper, we extend their definitions to MCpk and MCp for multivariate normal processes with flexible specification regions. To link the index to the yield, we establish a ‘reachable’ lower bound for the process yield as a function of MCpk. An algorithm suitable for such processes is developed to compute the natural estimate of MCpk from process data. Furthermore, we construct via the bootstrap approach the lower confidence bound of MCpk, a measure often used by producers for quality assurance to consumers. As for BCp, we first modify the original definition with a simple preprocessing step to make BCp scale-invariant. A very efficient algorithm is developed for computing a natural estimator of BCp. This new approach of BCp can be easily extended to MCp for multivariate processes. For BCp, we further derive an approximate normal distribution for , which enables us to construct procedures for making statistical inferences about process capability based on data, including the hypothesis testing, confidence interval, and lower confidence bound. Finally, the proposed procedures are demonstrated with three real data sets. Copyright © 2012 John Wiley & Sons, Ltd.