Effects of Parameter Estimation on the Multivariate Distribution-free Phase II Sign EWMA Chart

Multivariate nonparametric control charts can be very useful in practice and have recently drawn a lot of interest in the literature. Phase II distribution‐free (nonparametric) control charts are used when the parameters of the underlying unknown continuous distribution are unknown and can be estimated from a sufficiently large Phase I reference sample. While a number of recent studies have examined the in‐control (IC) robustness question related to the size of the reference sample for both univariate and multivariate normal theory (parametric) charts, in this paper, we study the effect of parameter estimation on the performance of the multivariate nonparametric sign exponentially weighted moving average (MSEWMA) chart. The in‐control average run‐length (ICARL) robustness and the out‐of‐control shift detection performance are both examined. It is observed that the required amount of the Phase I data can be very (perhaps impractically) high if one wants to use the control limits given for the known parameter case and maintain a nominal ICARL, which can limit the implementation of these useful charts in practice. To remedy this situation, using simulations, we obtain the ‘corrected for estimation’ control limits that achieve a desired nominal ICARL value when parameters are estimated for a given set of Phase I data. The out‐of‐control performance of the MSEWMA chart with the correct control limits is also studied. The use of the corrected control limits with specific amounts of available reference sample is recommended. Otherwise, the performance the MSEWMA chart may be seriously affected under parameter estimation.