Abstract

Since parameter estimation degrades chart performance, it is important to design a control chart correctly, that is, taking account of the estimation effects. To this end, two perspectives are available in the literature: the unconditional, which focuses on the unconditional in-control (IC) average run length (${\mathit{ARL}}_0$), and the conditional, which focuses on the IC run-length distribution conditioned on the parameter estimates and the exceedance probability criterion (EPC). Much of the literature on this topic is in the context of monitoring the mean. However, monitoring the variance is important in the larger monitoring context, not only per se, but also because a reliable and stable estimate of the process variance is required in the first place for setting up the control chart for the mean. With this in mind, and given that a recent paper studied the design of the $\overline{X}$ chart, here we consider the S² chart and examine the effects of each perspective on the design and IC performance. To this end, we first compare the required number of Phase I samples and the control limit adjustments in two cases: the upper one-sided chart and the equal-tailed two-sided chart. Second, we examine the performance of each chart, designed according to one perspective, under the other perspective. Results show major differences in the impact and consequences of the adopted chart design perspective on chart performance. An illustration with a real dataset is provided. Finally, an overall summary and some conclusions are presented.

摘要

由于参数估计会降低图表性能，因此正确设计控制图非常重要，即考虑到估计效果。为此，文献中有两种观点：无条件的，侧重于无条件的控制（IC）平均运行长度（${\mathit{ARL}}_0$) 和有条件的，它侧重于以参数估计和超出概率标准 (EPC) 为条件的 IC 运行长度分布。许多关于这个主题的文献都是在监测平均值的背景下进行的。然而，在更大的监测环境中监测方差很重要，不仅本身，而且因为首先需要对过程方差进行可靠和稳定的估计，以建立均值控制图。考虑到这一点，并鉴于最近的一篇论文研究了$\overline{X}$图表，这里我们考虑S ²图表并检查每个角度对设计和 IC 性能的影响。为此，我们首先比较了上单面图和等尾双面图两种情况下所需的第一阶段样本数和控制限调整。其次，我们检查每张图表的表现，根据一个角度设计，在另一个角度下。结果表明，采用的图表设计视角对图表性能的影响和后果存在重大差异。提供了带有真实数据集的插图。最后，给出了一个总体总结和一些结论。