Comparison of joint control schemes for multivariate normal i.i.d. output

Abstract

The performance of a product frequently relies on more than one quality characteristic. In such a setting, joint control schemes are used to determine whether or not we are in the presence of unfavorable disruptions in the location ( $${\varvec{\mu }}$$ ) and spread ( $${\varvec{\varSigma }}$$ ) of a vector of quality characteristics. A common joint scheme for multivariate output comprises two charts: one for $${\varvec{\mu }}$$ based on a weighted Mahalanobis distance between the vector of sample means and the target mean vector; another one for $${\varvec{\varSigma }}$$ depending on the ratio between the determinants of the sample covariance matrix and the target covariance matrix. Since we are well aware that there are plenty of quality control practitioners who are still reluctant to use sophisticated control statistics, this paper tackles Shewhart-type charts for $${\varvec{\mu }}$$ and $${\varvec{\varSigma }}$$ based on three pairs of control statistics depending on the nominal mean vector and covariance matrix, $${\varvec{\mu }}_0$$ and $${\varvec{\varSigma }}_0$$ . We either capitalize on existing results or derive the joint probability density functions of these pairs of control statistics in order to assess the ability of the associated joint schemes to detect shifts in $${\varvec{\mu }}$$ or $${\varvec{\varSigma }}$$ for various out-of-control scenarios. A comparison study relying on extensive numerical and simulation results leads to the conclusion that none of the three joints schemes for $${\varvec{\mu }}$$ and $${\varvec{\varSigma }}$$ is uniformly better than the others. However, those results also suggest that the joint scheme with the control statistics $$n \, ( \bar{\mathbf {X}}-{\varvec{\mu }}_0 )^\top \, {\varvec{\varSigma }}_0^{-1} \, ( \bar{\mathbf {X}}-{\varvec{\mu }}_0 )$$ and $$\hbox {det} \left( (n-1) \mathbf{S} \right) / \hbox {det} \left( {\varvec{\varSigma }}_0 \right) $$ has the best overall average run length performance.