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

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

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 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 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

Guaranteed Conditional Performance of Control Charts via Bootstrap Methods

To use control charts in practice, the in-control state usually has to be estimated. This estimation has a detrimental effect on the performance of control charts, which is often measured for example by the false alarm probability or the average run length. We suggest an adjustment of the monitoring schemes to overcome these problems. It guarantees, with a certain probability, a conditional performance given the estimated in-control state. The suggested method is based on bootstrapping the data used to estimate the in-control state. The method applies to different types of control charts, and also works with charts based on regression models, survival models, etc. If a nonparametric bootstrap is used, the method is robust to model errors. We show large sample properties of the adjustment. The usefulness of our approach is demonstrated through simulation studies.Comment: 21 pages, 5 figure

Distribution-free double-sampling precedence monitoring scheme to detect unknown shifts in the location parameter

In most applications, parametric monitoring schemes are used to monitor the majority of industrial and nonindustrial processes in order to improve the quality of the outputs or services. However, parametric monitoring schemes are known to underperform when the normality assumption is not met or when there is not enough information about the symmetry or asymmetry nature of the process underlying distribution. Hence, in this paper, a new nonparametric Phase II Shewhart-type double-sampling (DS) monitoring scheme based on the precedence statistic is proposed in order to efficiently monitor quality processes when the underlying process distribution departs from normality. The performance is investigated using the average run length (ARL), standard deviation of the run length (SDRL), expected ARL (EARL) and expected average number of observations to signal (EANOS), and the average sample sizes (ASS) metrics. The latter metrics are computed using Monte Carlo simulation and exact formulae. In general, it is shown that the new DS precedence scheme outperforms the existing basic Shewhart precedence scheme with and without supplementary runs rules in many situations. A real-life illustrative example based on a filling process of milk bottles is provided to demonstrate the application and implementation of the new DS precedence monitoring scheme.https://wileyonlinelibrary.com/journal/qre2022-06-18hj2022Statistic

A class of \v{S}id\'ak-type tests based on maximal precedence and exceedance statistic

A class of nonparametric two-sample tests has been proposed in this article. As a generalization of the original \v{S}id\'aks' test, the proposed test statistic is developed as the sum of the maximal precedence and maximal exceedance statistics. Unlike the \v{S}id\'ak-type precedence-exceedance test and the maximal precedence test, the proposed test is suitable for a two-sided alternative while being free from any parametric assumption. Exact distribution of the test statistic is obtained under the null as well as under the Lehmann alternative. Power value comparison has been carried out that shows the competency of the proposed test as a useful alternative to a number of existing tests based on precedence-exceedance statistics. Real-life example is provided to illustrate the application of the proposed test