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

  • Manuel Cabral MoraisEmail author
  • Wolfgang Schmid
  • Patrícia Ferreira Ramos
  • Taras Lazariv
  • António Pacheco
Original Paper


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.


Average run length Joint schemes Multivariate normal Statistical process control 



This work was partially supported by FCT (Fundação para a Ciência e a Tecnologia) through projects UID/Multi/04621/2013 and PEst-OE/MAT/UI0822/ 2014. We are most grateful to the two reviewers who selflessly devoted their time to scrutinize this work and offered very pertinent comments that led to a shorter and improved version of the original manuscript.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and CEMAT (Center for Computational and Stochastic Mathematics), Instituto Superior TécnicoULisboaLisbonPortugal
  2. 2.Department of StatisticsEuropean University ViadrinaFrankfurt (Oder)Germany
  3. 3.CEMATUniversidade Autónoma de LisboaLisbonPortugal
  4. 4.Center of Information Services and High Performance ComputingTU DresdenDresdenGermany

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