Research Article Published online 30 October 2009 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/qre.1079 Properties of the Exponential EWMA Chart with Parameter Estimation Guney Ozsan, a Murat Caner Testik b∗ † and Christian H. Weiß c Count rates may reach very low levels in production processes with low defect levels. In such settings, conventional control charts for counts may become ineffective since the occurrence of many samples with zero defects would cause control statistic to be consistently zero. Consequently, the exponentially weighted moving average (EWMA) control chart to monitor the time between successive events (TBE) or counts has been introduced as an effective approach for monitoring processes with low defect levels. When the counts occur according to a Poisson distribution, the TBE observations are distributed as exponential. Although the assumption of exponential distribution is a reasonable choice as a model of TBE observations, its parameter, i.e. the mean (also the standard deviation), is rarely known in practice and its estimate is used in place of the unknown parameter when constructing the exponential EWMA chart. In this article, we investigate the effects of parameter estimation on the performance measures (average run length, standard deviation, and percentiles of the run length distribution) of the exponential EWMA control chart. A comprehensive analysis of the conditional performance measures of the chart shows that the effect of estimation can be serious, especially if small samples are used. An investigation of the marginal performance measures, which are calculated by averaging the conditional performance measures over the distribution of the parameter estimator, allows us to provide explicit sample size recommendations in constructing these charts to reach a satisfactory performance in both the in-control and the out-of-control situation. Copyright © 2009 John Wiley & Sons, Ltd. Keywords: statistical process control; time between events (TBE); exponential EWMA control chart; estimated parameters; inverse Gamma distribution; low defect level 1. Introduction B enefits of statistical process control or specifically quality control charts can be extended to production environments with low defect levels. In general, control charts of count data such as the c-, cumulative sum (CUSUM), and exponentially weighted moving average (EWMA) are valuable tools in quality improvement activities 1--3 . Nevertheless, these control charts for monitoring the occurrence rate of events, e.g. the number of nonconforming items or defects, may underperform in providing information regarding the process status when the count rates are very low. In particular, counts of events in many fixed time intervals will be zero and a control chart of counts will have narrow control limits, which will cause frequent false alarms. Instead of monitoring the number of counts in fixed time intervals, Xie et al. 4 suggest the monitoring of cumulative counts of conforming items between two nonconforming items. Likewise, monitoring the quantity of product inspected to observe a defect, a so-called cumulative quantity control chart, is considered in Chan et al. 5 . Alternatively, monitoring of the time between consecutive events (TBE) rather than counts in fixed/ variable intervals has also been suggested by several authors (see, e.g. 6--9 ). Performance comparisons as well as implementations indicate the merits of TBE monitoring. A common model for the occurrence of rare events is the homogeneous Poisson process. That is, the number of events in a time period follows a Poisson distribution and the TBE are independent and identically distributed (i.i.d.) exponential random variables. Consequently, control charts utilizing the TBE statistic are generally designed under the exponential model for TBE observations 3--9 . Furthermore, since the statistics TBE and the counts of events are related for a homogeneous Poisson process, monitoring the mean of the exponential TBE is an alternative to monitoring the rate of the Poisson counts of events at fixed time intervals. Note that a decrease in the mean of the TBE, and correspondingly an increase in the rate of counts in an interval indicates a process quality deterioration and vice versa. To effectively monitor a homogeneous Poisson process, extensions to a Department of Industrial Engineering, Middle East Technical University, Ankara 06531, Turkey b Department of Industrial Engineering, Hacettepe University, Ankara 06800, Turkey c Department of Mathematics, Darmstadt University of Technology, Darmstadt, Germany ∗ Correspondence to: Murat Caner Testik, Faculty of Engineering, Department of Industrial Engineering, Hacettepe University, 06800 Beytepe Ankara, Turkey. † E-mail: mtestik@hacettepe.edu.tr Contract/grant sponsor: Hacettepe University’s Scientific Research Fund BAP; contract/grant number: 0601602006 Copyright © 2009 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2010, 26 555--569 555