Incorporating parameter uncertainty into the setup of EWMA control charts monitoring normal variance

Most of the literature concerned with the design of control charts relies on perfect knowledge of the distribution for at least the good (so-called in-control) process. Some papers treated the handling of EWMA charts monitoring normal mean in case of unknown parameters - refer to Jones, Champ and Rigdon (2001) for a good introduction. In Jensen, Jones-Farmer, Champ, and Woodall (2006): “Effects of Parameter Estimation on Control Chart Properties: A Literature Review” a nice overview was given. Additionally, it was mentioned that it would be interesting and useful to evaluate and take into account these effects also for variance control charts. Here, we consider EWMA charts for monitoring the normal variance. Given a sequence of batches of size $n$, $\{X_{i j}\}$, $i=1,2,\ldots$ and $j=1,2,\ldots,n$ utilize the following EWMA control chart: \begin{align*} Z_0 & = z_0 = \sigma_0^2 = 1 \,, \\ Z_i & = (1-\lambda) Z_{i-1} + \lambda S_i^2 \,,\; i = 1,2,\ldots \,,\\ & \qquad\qquad S_i^2 = \frac{1}{n-1} \sum_{i=1}^n (X_{ij} - \bar X_i)^2 \,,\; \bar X_i = \frac{1}{n} \sum_{i=1}^n X_{ij} \,, \\ L & = \inf \left\{ i \in I\!\!N: Z_i > c_u \sigma_0^2 \right\} \,. \end{align*} The parameters $\lambda \in (0,1]$ and $c_u \gt 0$ are chosen to enable a certain useful detection performance (not too much false alarms and quick detection of changes). The most popular performance measure is the so-called Average Run Length (ARL), that is $E_{\sigma}(L)$ for the true standard deviation $\sigma$. If $\sigma_0$ has to be estimated by sampling data during a pre-run phase, then this uncertain parameter effects, of course, the behavior of the applied control chart. Typically the ARL is increased. Most of the papers about characterizing the uncertainty impact deal with the changed ARL patterns and possible adjustments. Here, a different way of designing the chart is treated: Setup the chart through specifying a certain false alarm probability such as $P_{\sigma_0}(L\le 1000) \le \alpha$. This results in a specific $c_u$. Here we describe a feasible way to determine this value $c_u$ also in case of unknown parameters for a pre-run series of given size (and structure). A two-sided version of the introduced EWMA scheme is analyzed as well.