Creating control charts with heteroscedastic variable I want to make a Shewhart chart with a charting statistic that is symmetrical but heavy-tailed (e.g., Cauchy). Does there exist anything like a Z-score with a heteroscedastic variance model? For instance,
\begin{equation}
Z = \frac{x - x_0}{ \sigma(x, \theta)}
\end{equation}
where $\theta$ is a hyperparameter on which the variance depends [and $\sigma$ might be estimated from ($x, \theta$)]. The other option might be to just partition the data $x$ according to ($x, \theta$) and create control charts for each subgroup ($x$, $\theta$).
The appeal of these approaches is that for sub-domains of $\theta$, $x$ is approximately normally distributed so typical control limits for the Z-score could then be used. I have not seen this in the literature but was wondering if it might be a valid method, or is there something better?
 A: I do not recall seeing anything specifically like this in the literature either.  There si a solution for calculationg $C_p$ and $C_{pk}$ in the literature that may be helpful here.
The National Institute of Science and Technology recommends
the use of $C_{npk}$, the non-parametric $C_{pk}$, for nonnormal
data.  This statistic is defined as $$C_{npk}=\min \left\{ \frac{USL-\tilde{x}}{X_{99.5\%}-\tilde{x}},\frac{\tilde{x}-LSL}{\tilde{x}-X_{.5\%}}\right\}$$ Where $X_{99.5\%}$, $\tilde{x}$, and $X_{.5\%}$ are the 99.5th, 50th, and .5th percentiles of the data within the desired distribution (e.g. Cauchy), respectively.
A Shewhart chart should then be easy to develop using a similar methodology where the basis is the $\tilde{x}-R$ chart.  The updates for the $\tilde{x}$ portion of the chart would be to use $\tilde{x}+X_{99.5\%}$ and $\tilde{x}-X_{.5\%}$ as the control limits.  The $R$ chart may pose some unexpected problems, however it may be robust enough to handle the ranges in each subgroup for the median.
