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.

中文翻译：

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多元法线输出的联合控制方案比较

产品的性能通常取决于多个质量特征。在这种情况下，联合控制方案可用来确定我们是否在位置（\（{\ varvec {\ mu}} \）\）和扩展（\（{\ varvec {\质量特征向量的varSigma}} \））。一种用于多变量输出的常见联合方案包括两个图表：一个基于样本均值向量与目标均值向量之间的加权Mahalanobis距离用于\（{\ varvec {\ mu}} \）；另一个用于\（{\ varvec {\ varSigma}} \）取决于样本协方差矩阵和目标协方差矩阵的行列式之间的比率。由于我们知道很多质量控制从业人员仍然不愿意使用复杂的控制统计信息，因此本文针对\（{\ varvec {\ mu}} \）和\（{\ varvec { \ varSigma}} \）基于标称平均向量和协方差矩阵\（{\ varvec {\ mu}} _ 0 \）和\（{\ varvec {\ varSigma {}。我们要么利用现有结果，要么派生这些控制统计对的联合概率密度函数，以评估相关联合方案检测能力变化的能力。\（{\ varvec {\ mu}} \）或\（{\ varvec {\ varSigma}} \\）用于各种失控场景。的比较研究依赖于广泛的数字和模拟结果导致这样的结论，没有任何三个关节方案为\（{\ varvec {\亩}} \）和\（{\ varvec {\ varSigma}} \）是均匀地更好比其他人。但是，这些结果还表明具有控制统计量\（n \，（\ bar {\ mathbf {X}}-{\ varvec {\ mu}} _ 0）^ \ top \，{\ varvec {\ varSigma}} _ 0 ^ {-1} \，（\ bar {\ mathbf {X}}-{\ varvec {\ mu}} _ 0} \）和\（\ hbox {det} \ left（（n-1） \ mathbf {S} \ right）/ \ hbox {det} \ left（{\ varvec {\ varSigma}} _ 0 \ right）\）具有最佳的总体平均游程性能。