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 Cp, Cpk, and so on. This paper discusses a relatively new development in MPCIs, namely, CG(u,v), which is a multivariate analogue of Cp(u,v)-the celebrated superstructure of univariate process capability indices . Some statistical properties of CG(u,v) are studied, particularly of CG(0,0), a member MPCI of the superstructure, which measures potential capability of a multivariate process. A threshold value of CG(0,0) is computed, and this can be considered as a logical cut-off for other member indices of CG(u,v) as well. The expression for the upper limit of the proportion of nonconformance is derived as a function of CG(0,0). Density plots of asymptotic distributions of four major member indices of CG(u,v), namely, CG(0,0), CG(1,0), CG(0,1), and CG(1,1), are made. Finally, a numerical example is discussed to supplement the theory developed in this paper.