Distributional and inferential properties of some new multivariate process capability indices for symmetric specification region

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.