Statistical quality control is used to improve performance of processes. Since most of the processes are multivariate in nature, multivariate process capability indices (MPCIs) have been developed by many researchers depending on the context. However, it is generally difficult to understand and calculate MPCIs, compared to their univariate counterparts like $C_p$, $C_{pk}$, and so on. This paper discusses a relatively new development in MPCIs, namely, $C_G(u,v)$, which is a multivariate analogue of $C_p(u,v)$—the celebrated superstructure of univariate process capability indices . Some statistical properties of $C_G(u,v)$ are studied, particularly of $C_G(0,0)$, a member MPCI of the superstructure, which measures potential capability of a multivariate process. A threshold value of $C_G(0,0)$ is computed, and this can be considered as a logical cut‐off for other member indices of $C_G(u,v)$ as well. The expression for the upper limit of the proportion of nonconformance is derived as a function of $C_G(0,0)$. Density plots of asymptotic distributions of four major member indices of $C_G(u,v)$, namely, $C_G(0,0)$, $C_G(1,0)$, $C_G(0,1)$, and $C_G(1,1)$, are made. Finally, a numerical example is discussed to supplement the theory developed in this paper.

中文翻译：

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对称规范区域中一些新的多元过程能力指数的分布和推论性质

统计质量控制用于改善流程的性能。由于大多数过程本质上都是多变量的，因此许多研究人员已根据具体情况开发了多过程能力指数（MPCI）。但是，与单变量对应的MPCI相比，通常很难理解和计算MPCI。$C_p$， $C_{pķ}$， 等等。本文讨论了MPCI的一个相对较新的发展，即$C_G（ü，v）$，它是的多元类似物 $C_p（ü，v）$-著名的单变量过程能力指数的上层建筑。的一些统计性质$C_G（ü，v）$ 被研究，特别是 $C_G（0，0）$，它是上层建筑的MPCI成员，用于测量多元过程的潜在能力。阈值$C_G（0，0）$ 计算得出，这可以视为其他成员索引的逻辑分界 $C_G（ü，v）$也一样 不合格比例上限的表达式是根据以下公式得出的：$C_G（0，0）$。四个主要成员指数的渐近分布的密度图$C_G（ü，v）$，即 $C_G（0，0）$， $C_G（\mathrm{1个}，0）$， $C_G（0，\mathrm{1个}）$， 和 $C_G（\mathrm{1个}，\mathrm{1个}）$，制成。最后，讨论了一个数值示例，以补充本文开发的理论。