A Distribution‐free Control Chart for the Joint Monitoring of Location and Scale

Traditional statistical process control for variables data often involves the use of a separate mean and a standard deviation chart. Several proposals have been published recently, where a single (combination) chart that is simpler and may have performance advantages, is used. The assumption of normality is crucial for the validity of these charts. In this article, a single distribution‐free Shewhart‐type chart is proposed for monitoring the location and the scale parameters of a continuous distribution when both of these parameters are unknown. The plotting statistic combines two popular nonparametric test statistics: the Wilcoxon rank sum test for location and the Ansari–Bradley test for scale. Being nonparametric, all in‐control properties of the proposed chart remain the same and known for all continuous distributions. Control limits are tabulated for implementation in practice. The in‐control and the out‐of‐control performance properties of the chart are investigated in simulation studies in terms of the mean, the standard deviation, the median, and some percentiles of the run length distribution. The influence of the reference sample size is examined. A numerical example is given for illustration. Summary and conclusions are offered. Copyright © 2011 John Wiley & Sons, Ltd.     

A New Distribution‐free Control Chart for Joint Monitoring of Unknown Location and Scale Parameters of Continuous Distributions

While the assumption of normality is required for the validity of most of the available control charts for joint monitoring of unknown location and scale parameters, we propose and study a distribution‐free Shewhart‐type chart based on the Cucconi 1 statistic, called the Shewhart‐Cucconi (SC) chart. We also propose a follow‐up diagnostic procedure useful to determine the type of shift the process may have undergone when the chart signals an out‐of‐control process. Control limits for the SC chart are tabulated for some typical nominal in‐control (IC) average run length (ARL) values; a large sample approximation to the control limit is provided which can be useful in practice. Performance of the SC chart is examined in a simulation study on the basis of the ARL, the standard deviation, the median and some percentiles of the run length distribution. Detailed comparisons with a competing distribution‐free chart, known as the Shewhart‐Lepage chart (see Mukherjee and Chakraborti 2 ) show that the SC chart performs just as well or better. The effect of estimation of parameters on the IC performance of the SC chart is studied by examining the influence of the size of the reference (Phase‐I) sample. A numerical example is given for illustration. Summary and conclusions are offered. Copyright © 2013 John Wiley & Sons, Ltd.     

Nonparametric cost‐minimized Shewhart‐type process monitoring with restricted false alarm probability

Traditional Duncan‐type models for cost‐efficient process monitoring often inflate type I error probability. Nevertheless, controlling the probability of type I error or false alarms is one of the key issues in sequential monitoring of specific process characteristics. To this end, researchers often recommend economic‐statistical designs. Such designs assign an upper bound on type I error probability to avoid excessive false alarms while achieving cost optimality. In the context of process monitoring, there is a plethora of research on parametric approaches of controlling type I error probability along with the cost optimization. In the nonparametric setup, most of the existing works on process monitoring address one of the two issues but not both simultaneously. In this article, we present two distribution‐free cost‐efficient Shewhart‐type schemes for sequentially monitoring process location with restricted false alarm probability, based, respectively, on the sign and Wilcoxon rank‐sum statistics. We consider the one‐sided shift in location parameter in an unknown continuous univariate process. Nevertheless, one can easily extend our proposed schemes to monitor the two‐sided process shifts. We evaluate and compare the actual performance of the two monitoring schemes employing extensive computer simulation based on Monte Carlo. We investigate the effects of the size of the reference sample and the false alarm constraint. Finally, we provide two illustrative examples, each based on a realistic situation in the industry.     

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 ‐chart, and the nonparametric VSI sign chart. Results show that the VSI signed‐rank chart often performs favourably and should be used.     

A class of new nonparametric circular-grid charts for signal classification

In this paper, some efficient monitoring and post-signal follow-up approaches are studied and compared for joint surveillance of location and scale of a process using the notions of circular-grid (CG) schemes. Precisely, three variants of CG Cucconi schemes are introduced and compared with three variants of percentile modified Lepage (PML) schemes. One of the PML schemes is equivalent to the traditional CG Lepage scheme, while another may be viewed as the Lepage type statistic using Gastwrith score, which is also a powerful tool for process surveillance. Overall, one of the proposed CG Cucconi schemes is most effective in identifying a class of signals, whether it is a location shift or scale shift or a shift in both parameters. It also indicates the direction of the shifts in either or both the parameters. Detecting a downward scale shift is the most challenging task in joint monitoring, and to this end, a new bias-corrected CG Lepage scheme is introduced. We compare the competing schemes in terms of correct signal classification probabilities. We illustrate the use of the proposed schemes in monitoring the trip-duration data in cab services. Some concluding remarks and future research problems are offered.     

Optimal design of a distribution-free quality control scheme for cost-efficient monitoring of unknown location

Traditionally, a cost-efficient control chart for monitoring product quality characteristic is designed using prior knowledge regarding the process distribution. In practice, however, the functional form of the underlying process distribution is rarely known a priori. Therefore, the nonparametric (distribution-free) charts have gained more attention in the recent years. These nonparametric schemes are statistically designed either with a fixed in-control average run length or a fixed false alarm rate. Robust and cost-efficient designs of nonparametric control charts especially when the true process location parameter is unknown are not adequately addressed in literature. For this purpose, we develop an economically designed nonparametric control chart for monitoring unknown location parameter. This work is based on the Wilcoxon rank sum (hereafter WRS) statistic. Some exact and approximate procedures for evaluation of the optimal design parameters are extensively discussed. Simulation results show that overall performance of the exact procedure based on bootstrapping is highly encouraging and robust for various continuous distributions. An approximate and simplified procedure may be used in some situations. We offer some illustration and concluding remarks.     

Comparisons of Shewhart-type rank based control charts for monitoring location parameters of univariate processes

The nonparametric (distribution-free) control charts are robust alternatives to the conventional parametric control charts when the form of underlying process distribution is unknown or complicated. In this paper, we consider two new nonparametric control charts based on the Hogg–Fisher–Randle (HFR) statistic and the Savage rank statistic. These are popular statistics for testing location shifts, especially in right-skewed densities. Nevertheless, the control charts based on these statistics are not studied in quality control literature. In the current context, we study phase-II Shewhart-type charts based on the HFR and Savage statistics. We compare these charts with the Wilcoxon rank-sum chart in terms of false alarm rate, out-of-control average run-length and other run length properties. Implementation procedures and some illustrations of these charts are also provided. Numerical results based on Monte Carlo analysis show that the new charts are superior to the Wilcoxon rank-sum chart for a class of non-normal distributions in detecting location shift. New charts also provide better control over false alarm when reference sample size is small.     

Design and comparison of some Shewhart‐type schemes for simultaneous monitoring of Weibull parameters

Weibull distribution is one of the most important probability models used in modeling time between events, system reliability, and particle sizes, among others. Therefore, efficiently and consequently monitoring certain changes in Weibull process is considered as an important research topic. Various statistical process monitoring schemes have been developed for monitoring different process parameters, including some for Weibull parameters. Most of these schemes are, however, designed to monitor and control a single process parameter, although there are two important model parameters for Weibull distribution. Recently, several researchers studied various schemes for jointly monitoring the mean and variance of a normally distributed process using a single plotting statistic. Nevertheless, there is still dearth of researches in joint monitoring of non‐normal process parameters. In this context, we develop some control schemes for simultaneously monitoring the scale and shape parameters of processes that follow the Weibull distribution. Implementation procedures are developed, and performance properties of various proposed schemes are investigated. We also offer an illustrative example along with a summary and recommendations.     

Detecting changes in location using distribution‐free control charts with big data

This paper proposes a simple distribution‐free control chart for monitoring shifts in location when the process distribution is continuous but unknown. In particular, we are concerned with big data applications where there are sufficient in‐control data that can be used to specify certain quantiles of interest which, in turn, are used to assess whether the new, incoming data to be monitored are in control. The distribution‐free chart is shown to lose very little power against the Shewhart charts designed for normally distributed data. The proposed charts offer a practical and robust alternative to the classical Shewhart charts which assume normality, particularly when monitoring quantiles and the data distribution is skewed. The effect of the size of the reference sample is examined on the assumption that the quantiles are known. Conclusions and recommendations are offered.     

A double generally weighted moving average exceedance control chart

Since the inception of control charts by W. A. Shewhart in the 1920s, they have been increasingly applied in various fields. The recent literature witnessed the development of a number of nonparametric (distribution‐free) charts as they provide a robust and efficient alternative when there is a lack of knowledge about the underlying process distribution. In order to monitor the process location, information regarding the in‐control (IC) process median is typically required. However, in practice, this information might not be available due to various reasons. To this end, a generalized type of nonparametric time‐weighted control chart labeled as the double generally weighted moving average (DGWMA) based on the exceedance statistic (EX) is proposed. The DGWMA‐EX chart includes many of the well‐known existing time‐weighted control charts as special or limiting cases for detecting a shift in the unknown location parameter of a continuous distribution. The DGWMA‐EX chart combines the better shift detection properties of a DGWMA chart with the robust IC performance of a nonparametric chart, by using all the information from the start until the most recent sample to decide if a process is IC or out‐of‐control. An extensive simulation study reveals that the proposed DGWMA‐EX chart, in many cases, outperforms its counterparts.     

Two economically optimized nonparametric schemes for monitoring process variability

Controlling and reducing process variability is an essential aspect for maintaining the product or service quality. Even though most practitioners believe that an increasing process variability is often a more severe concern than a shift in location, barely a few research paid attention to the cost-efficient monitoring of process variability. Some of the existing studies addressed the dispersion aspect, assuming that the quality characteristic is Gaussian. Non-normal and complex distributions are not uncommon in modern production processes, time to event processes, or processes involving service quality. Unfortunately, we find no literature on economically designed nonparametric (distribution-free) schemes for monitoring process variability. This article introduces two Shewhart-type cost-optimized nonparametric schemes for monitoring the variability of any unknown but continuous processes to fill the research gap. The proposed monitoring schemes are based on two popular two-sample rank statistics for differences in scale parameters, known as the Ansari–Bradley statistic and the Mood statistic. We assess their actual performance for a set of process scenarios and illustrate the design along with the implementation steps. We discuss a practical problem related to product quality management. It is expected that the proposed schemes will be beneficial in various industrial operations.     

A combination of max‐type and distance based schemes for simultaneous monitoring of time between events and event magnitudes

Traditionally, two isolated sequential stopping rules are employed for monitoring the time of occurrence of an event (T) and the magnitude of an event (X) . Recently, several researchers recommend monitoring T and X together using some unified approach. A unified approach based on combinations of two statistics, one for monitoring T and the other for X , is often more efficient. Likewise, a new approach of simultaneous monitoring of location and scale parameters of a process, combining a max and a distance based statistics, is recently introduced in literature. Motivated by such emerging concepts, we design a new scheme combining a Max‐type and a Distance‐type schemes, referred to as the MT scheme, to monitor T  and X simultaneously and efficiently. It retains the advantages of both the Max‐type and the Distance‐type schemes for joint inference. The proposed scheme is very competent in detecting a shift in the process distribution of T  or X or both. Moreover, it is computationally simpler. It has nice exact expressions for design parameters. Therefore, it is easier to implement. It has a distinct advantage over its traditional counterparts in detecting moderate to large shifts. Finally, we illustrate the implementation of the proposed scheme with a real dataset of damage caused by outbreak of fire disaster.     

An Efficient Nonparametric EWMA Wilcoxon Signed‐Rank Chart for Monitoring Location

The statistical performance of traditional control charts for monitoring the process shifts is doubtful if the underlying process will not follow a normal distribution. So, in this situation, the use of a nonparametric control charts is considered to be an efficient alternative. In this paper, a nonparametric exponentially weighted moving average (EWMA) control chart is developed based on Wilcoxon signed‐rank statistic using ranked set sampling. The average run length and some other associated characteristics were used as the performance evaluation of the proposed chart. A major advantage of the proposed nonparametric EWMA signed‐rank chart is the robustness of its in‐control run length distribution. Moreover, it has been observed that the proposed version of the EWMA signed‐rank chart using ranked set sampling shows better detection ability than some of the competing counterparts including EWMA sign chart, EWMA signed‐rank chart, and the usual EWMA control chart using simple random sampling scheme. An illustrative example is also provided for practical consideration. Copyright © 2016 John Wiley & Sons, Ltd.     

A new nonparametric double exponentially weighted moving average control chart

There are many practical situations where the underlying distribution of the quality characteristic either deviates from normality or it is unknown. In such cases, practitioners often make use of the nonparametric control charts. In this paper, a new nonparametric double exponentially weighted moving average control chart on the basis of the signed-rank statistic is proposed for monitoring the process location. Monte Carlo simulations are carried out to obtain the run length characteristics of the proposed chart. The performance comparison of the proposed chart with the existing parametric and nonparametric control charts is made by using various performance metrics of the run length distribution. The comparison showed the superiority of the suggested chart over its existing parametric and nonparametric counterparts. An illustrative example for the practical implementation of the proposed chart is also provided by using the industrial data set.     

New control charts for monitoring the Weibull percentiles under complete data and Type‐II censoring

In this paper, we propose 3 new control charts for monitoring the lower Weibull percentiles under complete data and Type‐II censoring. In transforming the Weibull distribution to the smallest extreme value distribution, Pascaul et al (2017) presented an exponentially weighted moving average (EWMA) control chart, hereafter referred to as EWMA‐SEV‐Q, based on a pivotal quantity conditioned on ancillary statistics. We extended their concept to construct a cumulative sum (CUSUM) control chart denoted by CUSUM‐SEV‐Q. We provide more insights of the statistical properties of the monitoring statistic. Additionally, in transforming a Weibull distribution to a standard normal distribution, we propose EWMA and CUSUM control charts, denoted as EWMA‐YP and CUSUM‐YP, respectively, based on a pivotal quantity for monitoring the Weibull percentiles with complete data. With complete data, the EWMA‐YP and CUSUM‐YP control charts perform better than the EWMA‐SEV‐Q and CUSUM‐SEV‐Q control charts in terms of average run length. In Type‐II censoring, the EWMA‐SEV‐Q chart is slightly better than the CUSUM‐SEV‐Q chart in terms of average run length. Two numerical examples are used to illustrate the applications of the proposed control charts.     

CRPS chart: Simultaneously monitoring location and scale under data‐rich environment

The detection performance of a conventional control chart is usually degraded by a large sample size as in Wang and Tsung. This paper proposes a new control chart under data‐rich environment. The proposed chart is based on the continuous ranked probability score and aims to simultaneously monitor the location and the scale parameters of any continuous process. We simulate different monitoring schemes with various shift patterns to examine the chart performance. Both in‐control and out‐of‐control performances are studied through simulation studies in terms of the mean, the standard deviation, the median, and some percentiles of the average run length distribution. Simulation results show that the proposed chart keeps a high sensitivity to shifts in location and/or scale without any distributional assumptions, and the outperformance improves, as the sample size becomes larger. Examples are given for illustration.     

Design and implementation issues for a class of distribution-free Phase II EWMA exceedance control charts

Distribution-free (nonparametric) control charts can play an essential role in process monitoring when there is dearth of information about the underlying distribution. In this paper, we study various aspects related to an efficient design and execution of a class of nonparametric Phase II exponentially weighted moving average (denoted by NPEWMA) charts based on exceedance statistics. The choice of the Phase I (reference) sample order statistic used in the design of the control chart is investigated. We use the exact time-varying control limits and the median run-length as the metric in an in-depth performance study. Based on the performance of the chart, we outline implementation strategies and make recommendations for selecting this order statistic from a practical point of view and provide illustrations with a data-set. We conclude with a summary and some remarks.     

A distribution-free CUSUM chart for joint monitoring of location and scale based on the combination of Wilcoxon and Mood statistics

We propose a distribution-free cumulative sum (CUSUM) chart for joint monitoring of location and scale based on a Lepage-type statistic that combines the Wilcoxon rank sum and the Mood statistics. Monte Carlo simulations were used to obtain control limits and examine the in-control and out-of-control performance of the new chart. A direct comparison of the new chart was made with the original CUSUM Lepage based on Wilcoxon rank sum and Ansari-Bradley statistics. The result is a more powerful chart in most of the considered scenarios and thus a more useful CUSUM chart. An example using real data illustrates how the proposed control chart can be implemented.     

A new non‐parametric CUSUM mean chart

Not all data in practice came from a process with normal distribution. When the process distribution is non‐normal or unknown, the commonly used Shewhart control charts are not suitable. In this paper, a new non‐parametric CUSUM Mean Chart is proposed to monitor the possible small mean shifts in the process. The sampling properties of the new monitoring statistics are examined and the average run lengths of the proposed chart are examined. Two numerical examples are used to illustrate the proposed chart and compare with the two existing charts, assuming normality and Beta distribution, respectively. The CUSUM Mean Chart showed better detection ability than those two charts in monitoring and detecting small process mean shifts. Copyright © 2010 John Wiley & Sons, Ltd.     

Monitoring the Shape Parameter of a Weibull Regression Model in Phase II Processes

In this paper, the interest is focused on monitoring profiles with Weibull distributed‐response and common shape parameter γ in phase II processes. The monitoring of such profiles is completely possible by taking the natural logarithm of the Weibull‐distributed response. This is equivalent to characterize the correspondent process by an extreme value linear regression model with common scale parameter σ = γ^?1. It was found out that from the monitoring of the common log‐scale parameter of the extreme value linear regression model, with the help of a simple scheme, it can be obtained important information about the deterioration of the entire process assuming the β coefficients as nuissance parameters that do not have to be known but stable. Control charts are based on the relative log‐likelihood ratio statistic defined for the log‐scale parameter of the log‐transformation of the Weibull‐distributed response and its respective signed square root. It was also found out that some existing adjustments are needed in order to improve the accuracy of using the distributional properties of the monitoring statistics for relatively small and moderate sample sizes. Simulation studies suggest that resulting charts have appealing properties and work fairly acceptable when non‐large enough samples are available at discrete sampling moments. Detection abilities of the studied corrected control schemes improve when sample size increases. Copyright © 2014 John Wiley & Sons, Ltd.