Distribution-free Phase II Mann–Whitney control charts with runs-rules

Abstract

The addition of runs-rules has been recommended to improve the performance of classical, normal theory Shewhart-type control charts, for detecting small to moderate size shifts. In this paper, we consider adding both standard and improved runs-rules to enhance the performance of the distribution-free Phase II Shewhart-type chart based on the well-known Mann-Whitney statistic proposed by Chakraborti and Van de Wiel [1]. Standard runs-rules are typically of the form w-of-(w+v) with w > 1 and v 0 and the improved runs-rules scheme is a combination of the classical 1-of-1 runs-rule and the w-of-(w+v) runs-rules. The improved scheme improves the performance of the charts in detecting larger shifts while maintaining its performance in detecting small to moderate shifts. The in-control and out-of-control performance of the proposed runs-rules enhanced distribution-free charts are examined through extensive simulations. It is seen that the proposed charts have attractive performance compared to some competing charts, and are better in many cases. An illustrative example is provided, along with a summary and conclusions.

One‐sided runs‐rules schemes to monitor autocorrelated time series data using a first‐order autoregressive model with skip sampling strategies

“…In an effort to increase the sensitivity of the Shewhart charts, supplementary runs‐rules have been incorporated to these charts to improve their performance in detection of small shifts; see for instance Koutras et al for earlier research work on these. For more recent research work on runs‐rules, see for instance, Abbasi et al, Zaman et al, Khoo et al, Malela‐Majika et al, Kumar et al, Rakitzis, Arshad et al, Shongwe and Graham, Tran, Maravelakis et al, Lee and Khoo, and Mehmood et al…”

Distribution‐free mixed cumulative sum‐exponentially weighted moving average control charts for detecting mean shifts

“…In this study, we use the ARL and SDRL properties. For more details on how to compute these metrics, see for example, Malela-Majika et al 8 and Li et al 5 A number of authors argued that if a control chart is designed based on a specific shift size of a shift, its performance would deteriorate when the actual size of a mean shift is significantly different from the assumed size (cf Ou et al 19 and Wu et al 20 ). Hence, they recommended using measures of the overall performance instead of the ARL to assess a monitoring scheme.…”

“…One of the most important properties of a nonparametric control chart is that the IC characteristics of the run-length distribution are the same across all continuous distributions. In other words, the IC run-length distribution does not depend on the underlying process distribution (cf Malela-Majika et al 8 ). This property is known as IC robustness of nonparametric control charts.…”

“…The classical mixed cumulative sum‐exponentially weighted moving average (MCE) scheme and its reversed version (ie, mixed exponentially weighted moving average‐cumulative sum [MEC] scheme) proposed by Zaman et al and Abbas et al, respectively, are mostly used to improve the sensitivity of the classical CUSUM and EWMA control charts in detecting small to moderate mean shifts when the assumption of normality is met. Ryan et al and many other authors have shown that the efficiency of classical (or parametric) control charts highly deteriorates under non‐normal underlying process distributions (cf Woodall and Montgomery; Li et al; Ozturk & Sun; Chakraborti et al; and Malela‐Majika et al). They showed that there is a need of more robust and flexible control charts that do not require the normality assumption.…”

“…The classical mixed cumulative sum-exponentially weighted moving average (MCE) scheme and its reversed version (ie, mixed exponentially weighted moving average-cumulative sum [MEC] scheme) proposed by Zaman et al 1 and Abbas et al, 2 respectively, are mostly used to improve the sensitivity of the classical CUSUM and EWMA control charts in detecting small to moderate mean shifts when the assumption of normality is met. Ryan et al 3 8 ). They showed that there is a need of more robust and flexible control charts that do not require the normality assumption.…”