CUSUM charts with controlled conditional performance under estimated parameters

We study the effect of the Phase I estimation error on the cumulative sum (CUSUM) chart. Impractically large amounts of Phase I data are needed to sufficiently reduce the variation in the in-control average run lengths (ARL) between practitioners. To reduce the effect of estimation error on the chart's performance we design the CUSUM chart such that the in-control ARL exceeds a desired value with a specified probability. This is achieved by adjusting the control limits using a bootstrap-based design technique. Such approach does affect the out-of-control performance of the chart; however, we find that this effect is relatively small.     

Reduction of the Effect of Estimation Error on In‐control Performance for Risk‐adjusted Bernoulli CUSUM Chart with Dynamic Probability Control Limits

The in‐control performance of any control chart is highly associated with the accuracy of estimation for the in‐control parameter(s). For the risk‐adjusted Bernoulli cumulative sum (CUSUM) chart with a constant control limit, it had been shown that the estimation error could have a substantial effect on the in‐control performance. In our study, we examine the effect of estimation error on the in‐control performance of the risk‐adjusted Bernoulli CUSUM chart with dynamic probability control limits (DPCLs). Our simulation results show that the in‐control performance of risk‐adjusted Bernoulli CUSUM chart with DPCLs is also affected by the estimation error. The most important factors affecting estimation error are the specified desired in‐control average run length, the Phase I sample size, and the adverse event rate. However, the effect of estimation error is uniformly smaller for the risk‐adjusted Bernoulli CUSUM chart with DPCLs than for the corresponding chart with a constant control limit under various realistic scenarios. In addition, we found a substantial reduction in the mean and variation of the standard deviation of the in‐control run length when DPCLs are used. Therefore, use of DPCLs has yet another advantage when designing a risk‐adjusted Bernoulli CUSUM chart. Copyright © 2016 John Wiley & Sons, Ltd.     

Properties of the Markov‐Dependent Attribute Control Chart with Estimated Parameters

The performance of attribute control charts that monitor Markov‐dependent data is usually evaluated under the assumption of known process parameters, that is, known values of a the probability an item is nonconforming given the previous item is conforming and b the probability an item is conforming given the previous item is nonconforming. In practice, these parameters are usually not known and are calculated from an in‐control Phase I‐data set. In this paper, a comparison of the in‐control ARL (average run length) properties of the attribute chart for Markov‐dependent data with known and estimated parameters is presented. The probability distribution of the estimators is developed and used to calculate the in‐control ARL and standard deviation of the run length of the chart with estimated parameters. For particular values of a and b, the in‐control ARL values of the charts with estimated parameters may be very different than those with known parameters. The size of the Phase‐I data set needed for charts with estimated parameters to exhibit the same in‐control ARL properties as those with known parameters may vary widely depending on the parameters of the process, but in general, large samples are needed to obtain accurate estimates. As the Phase‐I sample size increases, the in‐control ARL values of the charts with estimated parameters approach that of the known parameter case but not in a monotonic fashion as in the case of the X‐bar chart. Copyright © 2015 John Wiley & Sons, Ltd.     

An exact method for designing Shewhart and S2 control charts to guarantee in-control performance

The in-control performance of Shewhart and S² control charts with estimated in-control parameters has been evaluated by a number of authors. Results indicate that an unrealistically large amount of Phase I data is needed to have the desired in-control average run length (ARL) value in Phase II. To overcome this problem, it has been recommended that the control limits be adjusted based on a bootstrap method to guarantee that the in-control ARL is at least a specified value with a certain specified probability. In this article we present simple formulas using the assumption of normality to compute the control limits and therefore, users do not have to use the bootstrap method. The advantage of our proposed method is in its simplicity for users; additionally, the control chart constants do not depend on the Phase I sample data.     

The np Chart with Guaranteed In‐control Average Run Lengths

We evaluate the in‐control performance of the np‐control chart with estimated parameter conditional on the Phase I sample. We then apply the bootstrap method to adjust the control chart limits to guarantee the desired in‐control average run length (ARL₀) value in the monitoring stage. The adjusted limits ensure that the ARL₀ would take a value greater than the desired value (say, B ) with a certain specified probability, that is, Pr(ARL ₀ > B ) = 1 ? ρ . The results indicate that adjusting control limits is not always necessary. We present a method to design control charts such that in control and out of control run lengths are guaranteed with pre specified probabilities. This method is an improvement of the classical statistical design approach employing constraints on in control and out of control ARL because, with this approach, there is a substantial probability that the actual run length in control may be too small. In addition, using the ARL approach may result in an actual out of control run length that is too large. Some numerical examples illustrate the efficacy of this design method. Copyright © 2016 John Wiley & Sons, Ltd.     

A comparative study of memory‐type control charts based on robust scale estimators

Exponential weighted moving average and cumulative sum (CUSUM) control charts are well‐known tool for their effectiveness in detecting small and moderate changes in the process parameters. To detect both large and small shifts, a new control structure is often recommended, named as combined Shewhart‐CUSUM control chart, which combines the advantages of a Shewhart chart with the CUSUM chart. In this paper, we investigate 11 different standard deviation estimators with the structures of these 3 types of control charts for monitoring the process dispersion under normal and contaminated normal environments. By applying Monte Carlo simulations, we compare the performance of these memory charts depending on 4 factors: (1) standard deviation estimator, (2) parent environment, (3) chart type, and (4) change magnitude. Extensive simulations are used to compute and study the run length profiles of these memory charts, including the average, the standard deviation, the several percentiles, and the cumulative distribution function curves of the run length distribution. It turns out that there is a significant difference between the run length distribution of the memory chart with estimated parameters and the analogous case with known parameters, even using the adjusted control limits under normal environment, and the difference is more severe when contaminations are present. This difference is gradually diminished when a large number of Phase I samples is used under normality, but it is not true in the contaminated cases.     

Practitioner advice: ERLDAT–A web-based tool to design and analyze EWMA control charts

The performance of a control chart is completely characterized by its run length distribution. Quality practitioners usually do not have access to the run length distribution but rely on the average run length (ARL) to design and evaluate the performance of an exponentially weighted moving average (EWMA) control chart. This article presents a web-based tool that provides users easy access to the Phase 2 (online or monitoring phase) run length distribution for a two-sided EWMA control chart with known parameters. The web-based tool calculates the run length distribution, percentiles of the run length distribution, as well as the mean (ARL) and variance (VRL) of the run length distribution. Additional functionality of the web-based tool includes plotting the run length distribution functions, building tables of the quantiles of the run length distribution, finding the smoothing parameter (λ) for an EWMA control chart for fixed control limit that satisfies ARL, VRL or percentile performance, and finding the control chart limit (k) for an EWMA control chart that satisfies ARL, VRL, or percentile performance. This tool and these techniques enable quality practitioners to better design and evaluate EWMA control charts.     

The Performance of the Multivariate Adaptive Exponentially Weighted Moving Average Control Chart with Estimated Parameters

The multivariate adaptive exponentially weighted moving average control chart (MAEWMA) can detect shifts of different sizes while diminishing the inertia problem to a large extent. Although it has several advantages compared to various multivariate charts, previous literature has not considered its performance when the parameters are estimated. In this study, the performance of the MAEWMA chart with estimated parameters is studied while considering the practitioner‐to‐practitioner variation. This kind of variation occurs due to using different Phase I samples by different practitioners in estimating the unknown parameters. The simulation results in this paper show that estimating the parameters results in extensively excessive false alarms and as a result a large number of Phase I samples is needed to achieve the desired in‐control performance. Using small number of Phase I samples in estimating the parameters may result in an in‐control ARL distribution that almost completely lies below the desired value. To handle this problem, we strongly recommend the use of a bootstrap‐based algorithm to adjust the control limit of the MAEWMA chart. This algorithm enables practitioners to achieve, with a certain probability, an in‐control ARL that is greater than or equal to the desired value while using the available number of Phase I samples. Copyright © 2015 John Wiley & Sons, Ltd.     

On the design of control charts with guaranteed conditional performance under estimated parameters

When designing control charts the in-control parameters are unknown, so the control limits have to be estimated using a Phase I reference sample. To evaluate the in-control performance of control charts in the monitoring phase (Phase II), two performance indicators are most commonly used: the average run length (ARL) or the false alarm rate (FAR). However, these quantities will vary across practitioners due to the use of different reference samples in Phase I. This variation is small only for very large amounts of Phase I data, even when the actual distribution of the data is known. In practice, we do not know the distribution of the data, and it has to be estimated, along with its parameters. This means that we have to deal with model error when parametric models are used and stochastic error because we have to estimate the parameters. With these issues in mind, choices have to be made in order to control the performance of control charts. In this paper, we discuss some results with respect to the in-control guaranteed conditional performance of control charts with estimated parameters for parametric and nonparametric methods. We focus on Shewhart, exponentially weighted moving average (EWMA), and cumulative sum (CUSUM) control charts for monitoring the mean when parameters are estimated.     

Phase I control chart based on a kernel estimator of the quantile function

To measure the statistical performance of a control chart in Phase I applications, the in‐control average run length (ARL) is the most frequently used parameter. In typical start up situations, control limits must be computed without knowledge of the underlying distribution of the quality characteristic. Assumptions of an underlying normal distribution can increase the probability of false alarms when the underlying distribution is non‐normal, which can lead to unnecessary process adjustments. In this paper, a control chart based on a kernel estimator of the quantile function is proposed. Monte Carlo simulation was used to evaluate the in‐control ARL performance of this chart relative to that of the Shewhart individuals control chart. The results indicate that the proposed chart is more robust to deviations in the assumed underlying distribution (with respect to the in‐control ARL) and results in an alternative method of designing control charts for individual units. Copyright © 2011 John Wiley & Sons, Ltd.     

A double exponentially weighted moving average chart based on likelihood ratio for monitoring an inflated Pareto process

In this paper, we present a new chart called a likelihood ratio based double exponentially weighted moving average (LR_DEWMA) chart to monitor the shape parameter of the inflated Pareto process. Three other control charts such as the Shewhart type, the classical cumulative sum (CUSUM), and the likelihood ratio based EWMA (LR_EWMA) charts are also investigated. The performance of the control charts is evaluated by the average run length (ARL) and standard deviation of run lengths (SDRL) computed through the Monte Carlo simulation approach. Moreover, the median run length (MRL) and some other run length (RL) percentiles are also considered in some cases. Different charts have shown the best performance in different cases. In detecting smaller shifts, while the LR_DEWMA chart outperformed the other charts in terms of ARL and MRL, the CUSUM chart has shown the best performance in terms of SDRL and IQR of RLs. The application of the proposed control charts is illustrated using a chromatography analyses data from the food industry.     

Exponential CUSUM Charts with Estimated Control Limits

Exponential CUSUM charts are used in monitoring the occurrence rate of rare events because the interarrival times of events for homogeneous Poisson processes are independent and identically distributed exponential random variables. In these applications, it is assumed that the exponential parameter, i.e. the mean, is known or has been accurately estimated. However, in practice, the in‐control mean is typically unknown and must be estimated to construct the limits for the exponential CUSUM chart. In this article, we investigate the effect of parameter estimation on the run length properties of one‐sided lower exponential CUSUM charts. In addition, analyzing conditional performance measures shows that the effect of estimation error can be significant, affecting both the in‐control average run length and the quick detection of process deterioration. We also provide recommendations regarding phase I sample sizes. This sample size must be quite large for the in‐control chart performance to be close to that for the known parameter case. Finally, we provide an industrial example to highlight the practical implications of estimation error, and to offer advice to practitioners when constructing/analyzing a phase I sample. Copyright © 2013 John Wiley & Sons, Ltd.     

Guaranteed conditional design of the median chart with estimated parameters

A common assumption for most control charts is the fact that the process parameters are supposed to be known or accurately estimated from Phase I samples. But, in practice, this is not a realistic assumption and the process parameters are usually estimated from a very limited number of samples that, in addition, may contain some outliers. Recently, a median chart with estimated parameters has been proposed to overcome these issues and it has been investigated in terms of the unconditional Average Run Length (ARL). As this median chart with estimated parameters does not take the “Phase I between‐practitioners” variability into account, in this paper, we suggest to revisit it using the Standard Deviation of the ARL as a measure of performance. The results show that this Standard Deviation of the ARL–based median chart actually requires a much larger amount of Phase I data than previously recommended to sufficiently reduce the variation in the chart performance. Due to the practical limitation of the number of the Phase I data, the bootstrap method is recommended as a good alternative approach to define new dedicated control chart parameters.     

A Reevaluation of the Adaptive Exponentially Weighted Moving Average Control Chart When Parameters are Estimated

The performance of control charts can be adversely affected when based on parameter estimates instead of known in‐control parameters. Several studies have shown that a large number of phase I observations may be needed to achieve the desired in‐control statistical performance. However, practitioners use different phase I samples and thus different parameter estimates to construct their control limits. As a consequence, there would be in‐control average run length (ARL) variation between different practitioners. This kind of variation is important to consider when studying the performance of control charts with estimated parameters. Most of the previous literature has relied primarily on the expected value of the ARL (AARL) metric in studying the performance of control charts with estimated parameters. Some recent studies, however, considered the standard deviation of the ARL metric to study the performance of control charts. In this paper, the standard deviation of the ARL metric is used to study the in‐control and out‐of‐control performance of the adaptive exponentially weighted moving average (AEWMA) control chart. The performance of the AEWMA chart is then compared with that of the Shewhart and EWMA control charts. The simulation results show that the AEWMA chart might represent a good solution for practitioners to achieve a reasonable amount of ARL variation from the desired in‐control ARL performance. In addition, we apply a bootstrap‐based design approach that provides protection against frequent false alarms without deteriorating too much the out‐of‐control performance. Copyright © 2014 John Wiley & Sons, Ltd.     

The Effect of Estimation Error on Risk‐Adjusted Survival Time CUSUM Chart Performance

Research on risk‐adjusted control charts has gained great interest in healthcare settings. Based on monitored variables (binary outcome or survival times), risk‐adjusted cumulative sum (CUSUM) charts are divided into Bernoulli and survival time CUSUM charts. The effect of estimation error on control chart performance has been systematically studied for Bernoulli CUSUM but not for survival time CUSUM in continuous time. We investigate the effect of estimation error on the performance of risk‐adjusted survival time CUSUM scheme in continuous time with the cardiac surgery data. The impact is studied with the use of the median run lengths (medRLs) and the standard deviation (SD) of medRLs for different sample sizes, specified in‐control median run length, adverse event rate and patient variability. Results show that estimation error affects the performance of risk‐adjusted survival time CUSUM chart significantly and the performance is more sensitive to the specified in‐control median run length (medRL₀) and adverse event rate. To take the estimation error into account, the practitioners can bootstrap many samples from Phase I data and then determine the threshold that can guarantee at least a medRL₀ with certain probability under which false alarms occur less frequently and meanwhile out‐of‐control alarms don't signal too slow. Moreover, additional event occurrences can be used to update the estimation but should be from in‐control process. Finally, non‐parametric bootstrap can be applied to reduce model misspecification error. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal design of one-sided exponential cumulative sum charts with known and estimated parameters based on the median run length

As a useful tool in statistical process control (SPC), the exponential control chart is more and more popular for monitoring high-quality processes. Considering both known and estimated parameter cases, the one-sided exponential cumulative sum (CUSUM) charts are studied in this paper through a Markov chain approach. Because the shape of the run length (RL ) distribution of the one-sided exponential CUSUM charts is skewed and it also changes with the mean shift size and the number of Phase I samples used to estimate the process parameter, the median run length (MRL ) is employed as a good alternative performance measure for the charts. The optimal design procedures based on MRL of the one-sided exponential CUSUM charts with known and estimated parameters are discussed. By comparing the MRL performance of the chart with known parameters with the one of the chart with estimated parameters, we investigate the effect of estimated process parameters on the properties of the chart. Finally, an application is illustrated to show the implementation of the chart.     

Effects of Parameter Estimation on the Multivariate Distribution‐free Phase II Sign EWMA Chart

Multivariate nonparametric control charts can be very useful in practice and have recently drawn a lot of interest in the literature. Phase II distribution‐free (nonparametric) control charts are used when the parameters of the underlying unknown continuous distribution are unknown and can be estimated from a sufficiently large Phase I reference sample. While a number of recent studies have examined the in‐control (IC) robustness question related to the size of the reference sample for both univariate and multivariate normal theory (parametric) charts, in this paper, we study the effect of parameter estimation on the performance of the multivariate nonparametric sign exponentially weighted moving average (MSEWMA) chart. The in‐control average run‐length (ICARL) robustness and the out‐of‐control shift detection performance are both examined. It is observed that the required amount of the Phase I data can be very (perhaps impractically) high if one wants to use the control limits given for the known parameter case and maintain a nominal ICARL, which can limit the implementation of these useful charts in practice. To remedy this situation, using simulations, we obtain the “corrected for estimation” control limits that achieve a desired nominal ICARL value when parameters are estimated for a given set of Phase I data. The out‐of‐control performance of the MSEWMA chart with the correct control limits is also studied. The use of the corrected control limits with specific amounts of available reference sample is recommended. Otherwise, the performance the MSEWMA chart may be seriously affected under parameter estimation. Copyright © 2016 John Wiley & Sons, Ltd.     

A Multivariate Exponentially Weighted Moving Average Control Chart

A multivariate extension of the exponentially weighted moving average (EWMA) control chart is presented, and guidelines given for designing this easy-to-implement multivariate procedure. A comparison shows that the average run length (ARL) performance of this chart is similar to that of multivariate cumulative sum (CUSUM) control charts in detecting a shift in the mean vector of a multivariate normal distribution. As with the Hotelling's χ² and multivariate CUSUM charts, the ARL performance of the multivariate EWMA chart depends on the underlying mean vector and covariance matrix only through the value of the noncentrality parameter. Worst-case scenarios show that Hotelling's χ² charts should always be used in conjunction with multivariate CUSUM and EWMA charts to avoid potential inertia problems. Examples are given to illustrate the use of the proposed procedure.     

The Design of the ARL‐Unbiased S2 Chart When the In‐Control Variance Is Estimated

The S² chart has been known as a powerful tool to monitor the variability of the normal process. When the variance of the process is unknown, it needs to be estimated by Phase I samples. It is well known that there are serious effects of parameter estimation on the performance of the S² chart based on known parameter assumption. If the effects of parameter estimation are not considered, it can lead to an increase in the number of false alarms and a reduction in the ability of the chart to detect process changes except for very small shifts in the variance. Based on the criterion of average run length (ARL) unbiased, a S² control chart is developed when the in‐control variance is estimated. The performance of the proposed control chart is also evaluated in terms of the ARL and standard deviation of the run length. Finally, an example is used to illustrate the proposed control chart. Copyright © 2013 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.