Distributional Properties of Multivariate Process Capability Indices under Normal and Non-normal Distributions

Multivariate capability analysis has been the focus of study in recent years, during which many authors have proposed different multivariate capability indices. In the operative context, capability indices are used as measures of the ability of the process to operate according to specifications. Because the numerical value of the index is used to conclude about the capability of the process, it is essential to bear in mind that almost always that value is obtained from a sample of process units. Therefore, it is really necessary to know the properties that the indices have when they are calculated on sampling information, in order to assess the goodness of the inferences made from them. In this work, we conduct a simulation study to investigate distributional properties of two existing indices: NMCpm index based on ratio of volumes and Mp2 index based on principal component analysis. We analyze the relative bias and the mean square error of the estimators of the indices, and we also obtain their empirical distributions that are used to estimate the probability that the indices classify correctly a process as capable or as incapable. The results allow us to recommend the use of one of these indices, as it has shown better properties.