The Cusum Median Chart for Known and Estimated Parameters

Abstract

The usual CUSUM chart for the mean (CUSUM-X¯) is a chart used to quickly detect small to moderate shifts in a process. In presence of outliers, this chart is known to be more robust than other mean-based alternatives like the Shewhart mean chart but it is nevertheless affected by these unusual observations because the mean (X¯) itself is affected by the outliers. An outliers robust alternative to the CUSUM-X¯ chart is the CUSUM median (CUSUM-X˜) chart, because it takes advantage of the robust properties of the sample median. This chart has already been proposed by other researchers and compared with other alternative charts in terms of robustness, but its performance has only been investigated through simulations. Therefore, the goal of this paper is not to carry out a robustness analysis but to study the effect of parameter estimation in the performance of the chart. We study the performance of the CUSUM-X˜ chart using a Markov chain method for the computation of the distribution and the moments of the run length. Additionally, we examine the case of estimated parameters and we study the performance of the CUSUM-X˜ chart in this case. The run length performance of the CUSUM-X˜ chart with estimated parameters is also studied using a proper Markov chain technique. Conclusions and recommendations are also given.