Control charts for monitoring the median in non-negative asymmetric data

Control charts are commonly used for monitoring the mean of processes. However, there are practical applications in which asymmetric data are the standard. In these scenarios, the use of robust statistics, such as the median, is advantageous over the mean. Based on this, we propose an empirical control chart for monitoring the median of a wide class of distributions, known as the log-symmetric class. Closed-form estimators, which perform better than the maximum likelihood estimator, are considered. Simulation studies are carried out with the following objectives: to evaluate the in-control and the out-control average run length; to evaluate the behavior of the control limits; and to compare the proposed method with a naive method based on the asymptotic distribution of the three estimators. The results indicate that the proposed approach presents better in-control average run length than the naive method and better power of detection for negative shifts in the median. A practical use of the proposed approach is illustrated with a real engineering problem, followed by a goodness of fit based on AIC and BIC, considering the most common asymmetric distributions. We also perform a residual analysis with the chosen distribution to verify its fit. Finally, based on the chosen distribution the proposed method indicates that there is an out-of-control point in phase II, which is not detected by the naive approach. Therefore, showing a gain in using the proposed method.