Distribution-free control charts gained momentum in recent years as they are more efficient in detecting a shift when there is a lack of information regarding the underlying process distribution. However, a distribution-free control chart for monitoring the process location often requires information on the in-control process median. This is somewhat challenging because, in practice, any information on the location parameter might not be known in advance and estimation of the parameter is therefore required. In view of this, a time-weighted control chart, labelled as the Generally Weighted Moving Average (GWMA) exceedance (EX) chart (in short GWMA-EX chart), is proposed for detection of a shift in the unknown process location; this chart is based on exceedance statistic when there is no information available on the process distribution. An extensive performance analysis shows that the proposed GWMA-EX control chart is, in many cases, better than its contenders.
The false alarm probability (FAP) is the metric typically used to design and evaluate the performance of Phase I control charts. It is shown that in situations where the exact or the asymptotic joint p.d.f. of the standardized charting statistics follows a singular standard multivariate normal distribution with a common negative correlation, the FAP of some Shewhart-type Phase I charts for the mean can be expressed as a multiple integral of the joint p.d.f. Hence the required charting constants can be calculated (and the control chart can be implemented) by evaluating this integral. A table with the charting constants is provided for some popular choices of the nominal FAP (denoted FAP 0 ) and the number of Phase I samples, m. The proposed methodology is useful to unify some existing Phase I charts and is illustrated with two charts from the literature: the p chart for the fraction nonconforming and the [Formula: see text] chart for the mean. A summary and some conclusions are provided.
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 article 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 runs-rules 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.
Abstract A significant challenge in statistical process monitoring (SPM) is to find exact and closed-form expressions (CFEs) (i.e. formed with constants, variables and a finite set of essential functions connected by arithmetic operations and function composition) for the run-length properties such as the average run-length ( $$ARL$$ ARL ), the standard deviation of the run-length ( $$SDRL$$ SDRL ), and the percentiles of the run-length ( $$PRL$$ PRL ) of nonparametric monitoring schemes. Most of the properties of these schemes are usually evaluated using simulation techniques. Although simulation techniques are helpful when the expression for the run-length is complicated, their shortfall is that they require a high number of replications to reach reasonably accurate answers. Consequently, they take too much computational time compared to other methods, such as the Markov chain method or integration techniques, and even with many replications, the results are always affected by simulation error and may result in an inaccurate estimation. In this paper, closed-form expressions of the run-length properties for the nonparametric double sampling precedence monitoring scheme are derived and used to evaluate its ability to detect shifts in the location parameter. The computational times of the run-length properties for the CFE and the simulation approach are compared under different scenarios. It is found that the proposed approach requires less computational time compared to the simulation approach. Moreover, once derived, CFEs have the added advantage of ease of implementation, cutting off on complex convergence techniques. CFE's can also easily be built into mathematical software for ease of computation and may be recalled for further work.
Nonparametric control charts are considered for the median and other percentiles based on runs of sign statistics above and below the control limits. It is noted that the sign charts are advantageous in certain practical situations. Expressions for the run-length distributions are derived using Markov chain theory; several examples are given. The in-control (IC) and the out-of-control (OOC) performance of these charts are studied and compared to the existing nonparametric Wilcoxon signed-ranked charts of Chakraborti and Eryilmaz (2007 Chakraborti , S. , Eryilmaz , S. ( 2007 ). A nonparametric Shewhart-type signed-rank control chart based on runs . Commun. Statist. Simul. Computat. 36 ( 2 ): 335 – 356 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) under the normal, the double exponential and the Cauchy distributions, using the average run-length (ARL), the standard deviation of the run-length (SDRL), the false alarm rate (FAR) and some percentiles of the run-length, including the median run-length (MDRL). It is shown that the proposed "runs-rules enhanced" sign charts offer more practically desirable IC ARL (ARL 0) and FAR values and perform better for some heavy-tailed distributions. Some concluding remarks are offered.
The homogeneously weighted moving average (HWMA) chart is a recent control chart that has attracted the attention of many researchers in statistical process control (SPC). The HWMA statistic assigns a higher weight to the most recent sample, and the rest is divided equally between the previous samples. This weight structure makes the HWMA chart more sensitive to small shifts in the process parameters when running in zero-state mode. Many scholars have reported its superiority over the existing charts when the process runs in zero-state mode. However, several authors have criticized the HWMA chart by pointing out its poor performance in steady-state mode due to its weighting structure, which does not reportedly comply with the SPC standards. This paper reviews and discusses all research works on HWMA-related charts (i.e., 55 publications) and provides future research ideas and new directions.
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