Multivariate process capability indices (MPCIs) have been proposed to measure multivariate process capability in real‐world application over the past three decades. For the practitioner's point of view, the intention of this paper is to examine the performances and distributional properties of probability‐based MPCIs. Considering issues of construction of capability indices in multivariate setup and computation with performance, we found that probability‐based MPCIs are a proper generalization of univariate basic process capability indices (PCIs). In the beginning of this decade, computation of probability‐based indices was a difficult and time‐consuming task, but in the computer age statistics, computation of probability‐based MPCIs is simple and quick. Recent work on the performance of MPCI NMC_pm and distributional properties of its estimator reasonably recommended this index, for use in practical situations. To study distributional properties of natural estimators of probability‐based MPCIs and recommended index estimator, we conducted simulation study. Though natural estimators of probability‐based indices are negatively biased, they are better with respect to mean, relative bias, mean square error. Probability‐based MPCI MC_pm is better as compared with NMC_pm with respect to performance and as its estimator quality. Hence, in real‐world practice, we recommend probability‐based MPCIs as a multivariate analogue of basic PCIs.

中文翻译：

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基于概率的多元过程能力指标的性质

在过去的三十年中，已经提出了多元过程能力指数（MPCI）来测量实际应用中的多元过程能力。从实践者的角度来看，本文旨在研究基于概率的MPCI的性能和分布特性。考虑到多元设置和性能计算中能力指标的构建问题，我们发现基于概率的MPCI是单变量基本过程能力指标（PCI）的适当概括。在这十年的初期，基于概率的索引的计算是一项艰巨且耗时的任务，但是在计算机时代统计中，基于概率的MPCI的计算既简单又快速。有关MPCI NMC _pm性能的最新工作估计量的分布特性和分布属性合理地建议使用此索引，以用于实际情况。为了研究基于概率的MPCI的自然估计量和推荐指数估计量的分布特性，我们进行了模拟研究。尽管基于概率的指数的自然估计量存在负偏倚，但相对于均值，相对偏倚和均方误差而言，它们的效果更好。在性能和估计质量方面，基于概率的MPCI MC _pm比NMC _pm更好。因此，在实际操作中，我们建议将基于概率的MPCI作为基本PCI的多变量类似物。