Improved Phase I control charts for monitoring times between events

In many situations, the times between certain events are observed and monitored instead of the number of events particularly when the events occur rarely. In this case, it is common to assume that the times between events follow an exponential distribution. Control charts are one of the main tools of statistical process control and monitoring. Control charts are used in phase I to assist operating personnel in bringing the process into a state of statistical control. In this paper, phase I control charts are considered for the observations from an exponential distribution with an unknown mean. A simulation study is carried out to compare the in-control robustness and out-of-control performance of the proposed chart. It is seen that the proposed charts are considerably more in-control robust than two competing charts and have comparable out-of control propertiesSouth African Research Chairs Initiative (SARChI) and University of Pretoria, in South Africa.http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1099-16382016-06-30hb201

Nonparametric (distribution-free) control charts : an updated overview and some results

Control charts that are based on assumption(s) of a specific form for the underlying process distribution are referred to as parametric control charts. There are many applications where there is insufficient information to justify such assumption(s) and, consequently, control charting techniques with a minimal set of distributional assumption requirements are in high demand. To this end, nonparametric or distribution-free control charts have been proposed in recent years. The charts have stable in-control properties, are robust against outliers and can be surprisingly efficient in comparison with their parametric counterparts. Chakraborti and some of his colleagues provided review papers on nonparametric control charts in 2001, 2007 and 2011, respectively. These papers have been received with considerable interest and attention by the community. However, the literature on nonparametric statistical process/quality control/monitoring has grown exponentially and because of this rapid growth, an update is deemed necessary. In this article, we bring these reviews forward to 2017, discussing some of the latest developments in the area. Moreover, unlike the past reviews, which did not include the multivariate charts, here we review both univariate and multivariate nonparametric control charts. We end with some concluding remarks.https://www.tandfonline.com/loi/lqen20hj2020Science, Mathematics and Technology Educatio

Distribution-free Phase II CUSUM control chart for joint monitoring of location and scale

Mukherjee and Chakraborti1 proposed a single distribution-free (nonparametric) Shewhart-type chart based on the Lepage2 statistic for simultaneously monitoring both the location and the scale parameters of a continuous distribution when both of these parameters are unknown. In the present work, we consider a single distribution-free CUSUM chart, based on the Lepage2 statistic, referred to as the CUSUM-Lepage (denoted by CL) chart. The proposed chart is distribution-free (nonparametric) and therefore, the in control (denoted IC) properties of the chart remain invariant and known for all continuous distributions. Control limits are tabulated for implementation of the proposed chart in practice. The IC and out of control (denoted OOC) performance properties of the chart are investigated through simulation studies in terms of the average, the standard deviation, the median and some percentiles of the run length distribution. Detailed comparison with a competing Shewhart-type chart is presented. Several existing CUSUM charts are also considered in the performance comparison. The proposed CL chart is found to perform very well in the location-scale models. We also examine the effect of the choice of the reference value (k) of CUSUM chart on the performance of the CL chart. The proposed chart is illustrated with a real data set. Summary and conclusions are presented.http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1099-16382016-02-28hb201

Improved Shewhart-type runs-rules nonparametric sign charts

Runs-rules are typically incorporated in control charts to increase their sensitivity to detect small process shifts. However, a drawback of this approach is that runs-rules charts are unable to detect large shifts quickly. In this paper improved runs-rules are introduced to the nonparametric sign chart, to address this limitation. Improved runs-rules are incorporated to maintain sensitivity to small process shifts, while having the added ability to detect large shifts in the process more efficiently. Performance comparisons between sign charts with runsrules and sign charts with improved runs-rules illustrate that the improved runs-rules are superior in performance for large shifts in the process, while maintaining the same sensitivity in the detection of small shifts.http://www.tandfonline.com/loi/lsta20hb201

Design and implementation of CUSUM exceedance control charts for unknown location

Nonparametric control charts provide a robust alternative in practice when the form of the underlying distribution is unknown. Nonparametric CUSUM (NPCUSUM) charts blend the advantages of a CUSUM with that of a nonparametric chart in detecting small to moderate shifts. In this paper, we examine efficient design and implementation of Phase II NPCUSUM charts based on exceedance (EX) statistics, called the NPCUSUM-EX chart. We investigate the choice of the order statistic from the reference (Phase I) sample that defines the exceedance statistic. We see that choices other than the median, such as the 75th percentile, can yield improved performance of the chart in certain situations. Furthermore, observing certain shortcomings of the average run-length (ARL), we use the median run-length (MRL) as the performance metric. The NPCUSUM-EX chart is compared with the NPCUSUM-Rank chart proposed by Li et al. (2010) based on the popular Wilcoxon rank-sum statistic. We also study the choice of the reference value, k, of the CUSUM charts. An illustration with real data is provided.http://www.tandfonline.com/loi/tprs202016-01-31hb201

Distribution-free exponentially weighted moving average control charts for monitoring unknown location

Distribution-free (nonparametric) control charts provide a robust alternative to a data analyst when there is lack of knowledge about the underlying distribution. A two-sided nonparametric Phase II exponentially weighted moving average (EWMA) control chart, based on the exceedance statistics (EWMA-EX), is proposed for detecting a shift in the location parameter of a continuous distribution. The nonparametric EWMA chart combines the advantages of a nonparametric control chart (known and robust in-control performance) with the better shift detection properties of an EWMA chart. Guidance and recommendations are provided for practical implementation of the chart along with illustrative examples. A performance comparison is made with the traditional (normal theory) EWMA chart for subgroup averages and a recently proposed nonparametric EWMA chart based on the Wilcoxon-Mann- Whitney statistics. A summary and some concluding remarks are given.http://www.elsevier.com/locate/csdanf201

Distribution-free exceedance CUSUM control charts for location

Distribution-free (nonparametric) control charts can be useful to the quality practitioner when the underlying distribution is not known. A Phase II nonparametric CUSUM chart based on the exceedance statistics, called the exceedance CUSUM chart, is proposed here for detecting a shift in the unknown location parameter of a continuous distribution. The exceedance statistics can be more efficient than rank-based methods when the underlying distribution is heavy-tailed and/or right-skewed, which may be the case in some applications, particularly with certain lifetime data. Moreover, exceedance statistics can save testing time and resources as they can be applied as soon as a certain order statistic of the reference sample is available. Guidelines and recommendations are provided for the chart’s design parameters along with an illustrative example. The inand out-of-control performance of the chart are studied through extensive simulations on the basis of the average run-length (ARL), the standard deviation of run-length (SDRL), the median run-length (MDRL) and some percentiles of run-length. Further, a comparison with a number of existing control charts, including the parametric CUSUM X chart and a recent nonparametric CUSUM chart based on the Wilcoxon rank-sum statistic, called the rank-sum CUSUM chart, is made. It is seen that the exceedance CUSUM chart performs well in many cases and thus can be a useful alternative chart in practice. A summary and some concluding remarks are given.http://www.tandfonline.com/loi/lssp20hb201

Nonparametric signed‐rank control charts with variable sampling intervals

Variable sampling interval (VSI) charts have been proposed in the literature for normal theory (parametric) control charts and are known to provide performance enhancements. In the VSI setting, the time between monitored samples is allowed to vary depending on what is observed in the current sample. Nonparametric (distribution‐free) control charts have recently come to play an important role in statistical process control and monitoring. In this paper a nonparametric Shewhart‐type VSI control chart is considered for detecting changes in a specified location parameter. The proposed chart is based on the Wilcoxon signed‐rank statistic and is called the VSI signed‐rank chart. The VSI signed‐rank chart is compared with an existing fixed sampling interval signed‐rank chart, the parametric VSI X‐chart, and the nonparametric VSI sign chart. Results show that the VSI signed‐rank chart often performs favourably and should be used.The South African Research Chairs Initiative at the University of Pretoria and by the Department of Information Systems, Statistics and Management Science, University of Alabama. Marien Graham's research was also supported by the National Research Foundation (Thuthuka programme: TTK14061168807; grant number: 94102), SARCHI Award to the third author from the National Research Foundation.http://wileyonlinelibrary.com/journal/qre2018-12-21hj2018Statistic

Design of variance control charts with estimated parameters: A head to head comparison between two perspectives

Since parameter estimation degrades chart performance, it is important to design a control chart correctly, that is, taking account of the estimation effects. To this end, two perspectives are available in the literature: the unconditional, which focuses on the unconditional in-control (IC) average run length ((Formula presented.)), and the conditional, which focuses on the IC run-length distribution conditioned on the parameter estimates and the exceedance probability criterion (EPC). Much of the literature on this topic is in the context of monitoring the mean. However, monitoring the variance is important in the larger monitoring context, not only per se, but also because a reliable and stable estimate of the process variance is required in the first place for setting up the control chart for the mean. With this in mind, and given that a recent paper studied the design of the (Formula presented.) chart, here we consider the S 2 chart and examine the effects of each perspective on the design and IC performance. To this end, we first compare the required number of Phase I samples and the control limit adjustments in two cases: the upper one-sided chart and the equal-tailed two-sided chart. Second, we examine the performance of each chart, designed according to one perspective, under the other perspective. Results show major differences in the impact and consequences of the adopted chart design perspective on chart performance. An illustration with a real dataset is provided. Finally, an overall summary and some conclusions are presented

Robustness of the EWMA control chart for individual observations

The traditional exponentially weighted moving average (EWMA) chart is one of the most popular control charts used in practice today. The in-control robustness is the key to the proper design and implementation of any control chart, lack of which can render its out-of-control shift detection capability almost meaningless. To this end, Borror et al. [5] studied the performance of the traditional EWMA chart for the mean for i.i.d. data. We use a more extensive simulation study to further investigate the in-control robustness (to non-normality) of the three different EWMA designs studied by Borror et al. [5]. Our study includes a much wider collection of non-normal distributions including light- and heavy-tailed and symmetric and asymmetric bi-modal as well as the contaminated normal, which is particularly useful to study the effects of outliers. Also, we consider two separate cases: (i) when the process mean and standard deviation are both known and (ii) when they are both unknown and estimated from an in-control Phase I sample. In addition, unlike in the study done by Borror et al. [5], the average run-length (ARL) is not used as the sole performance measure in our study, we consider the standard deviation of the run-length (SDRL), the median run-length (MDRL), and the first and the third quartiles as well as the first and the 99th percentiles of the in-control run-length distribution for a better overall assessment of the traditional EWMA chart’s in-control performance. Our findings sound a cautionary note to the (over) use of the EWMA chart in practice, at least with some types of non-normal data. A summary and recommendations are provided.STATOMET and the Department of Statistics at the University of Pretoria.http://www.tandfonline.com/loi/cjas20nf201