In-control Performance Research Articles

The Poisson INAR(1) one-sided EWMA chart with estimated parameters

The Poisson INAR(1) one-sided exponentially weighted moving average (EWMA) chart has been proposed to monitor integer-valued autoregressive processes of order 1 with a Poisson marginal distribution. It is common to assume that the INAR(1) process parameters are known or can be accurately estimated. However, in practice, the in-control process mean and autocorrelation coefficient are typically unknown and must be estimated. In this article, we investigate the effect of parameter estimation on the run length properties of the Poisson INAR(1) one-sided EWMA chart with the use of bivariate Markov chain approach. It is shown from the conditional in-control and out-of-control average run length values with different design parameters under various shift magnitudes that the effect of parameter estimation error can be significant. The effect due to the process mean estimation error is stronger than that due to the autocorrelation coefficient. Moreover, practitioners should rarely expect the in-control performance to be close to that obtained under the assumption that process parameters are known. We also provide sample size-recommendations regarding marginal performance. This sample size depends on the different shift magnitudes.

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 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) 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.

An Empirical-Likelihood-Based Multivariate EWMA Control Scheme

Nonparametric control charts are useful in statistical process control (SPC) when there is a lack of or limited knowledge about the underlying process distribution, especially when the process measurement is multivariate. This article develops a new multivariate SPC methodology for monitoring location parameter based on adapting a well-known nonparametric method, empirical likelihood (EL), to on-line sequential monitoring. The weighted version of EL ratio test is used to formulate the charting statistic by incorporating the exponentially weighted moving average control (EWMA) scheme, which results in a nonparametric counterpart of the classical multivariate EWMA (MEWMA). Some theoretical and numerical studies show that benefiting from using EL, the proposed chart possesses some favorable features. First, it is a data-driven scheme and thus is more robust to various multivariate non-normal data than the MEWMA chart under the in-control (IC) situation. Second, it is transformation-invariant and avoids the estimation of covariance matrix from the historical data by studentizing internally, and hence its IC performance is less deteriorated when the number of reference sample is small. Third, in comparison with the existing approaches, it is more efficient in detecting small and moderate shifts for multivariate non-normal process.

Hotelling's T<SUP align="right">2</SUP> charts with variable sampling interval and warning limit

The idea of variable warning limit is applied to variable sampling interval (VSI) Hotelling’s T2 charts. The purpose is to reduce the average frequency of switches between the sampling interval lengths of VSI T2 charts during their implementation. Expressions for the performance measures for variable sampling intervaland warning limit (VSIWL) T2 charts are developed. The performances of VSIWL T2 charts are compared numerically with that of static, VSI, and VSI (1, 3) T2 charts, where VSI (1, 3) T2 charts are the VSI T2 charts with runs rule (1, 3) for switching between sampling interval lengths. A VSIWL T2 chart yields different out-of-control performances for the same in-control statistical performance depending on the choices of values of its warning limit. The values of its warning limit can be chosen to yield the average frequency of switches between its sampling interval lengths dramatically and significantly smaller than that of VSI and VSI (1, 3) T2 charts, respectively, and to yield slightly bette...

A GLR Control Chart for Monitoring the Mean Vector of a Multivariate Normal Process

This paper develops a statistical process control (SPC) chart based on a generalized likelihood ratio (GLR) statistic to monitor the mean vector of a multivariate normal process. The performance of the GLR chart is compared with the performance of the Hotelling χ2 chart, the multivariate exponentially weighted moving average (MEWMA) chart, and a multi-MEWMA combination. Results show that the Hotelling χ2 chart and the MEWMA chart are only effective for a small range of shift sizes in the mean vector, while the GLR chart and some carefully designed multi-MEWMA combinations can give similarly better overall performance in detecting a wide range of shift magnitudes. Unlike most of these other options, the GLR chart does not require specification of tuning-parameter values by the user. The GLR chart also has the advantage in process diagnostics: at the time of a signal, estimates of change-point and out-of-control mean vector are immediately available to the user. All these advantages of the GLR chart make it a favorable option for practitioners. For the design of the GLR chart, a series of easy-to-use equations are provided to users for calculating the control limit to achieve a desired in-control performance.

A Directional Multivariate Control Chart for Monitoring Univariate Autocorrelated Processes

A new nonparametric multivariate control chart, based on a spatial-sign test and integrating the directional information from processes with the exponentially weighted moving average (EWMA) scheme, is developed for monitoring the mean of a univariate autocorrelated process. Simulation studies show that it has robustness in in-control (IC) performance, and it is more sensitive to the small and moderate mean shifts for non-normality underlying process than other existing multivariate chart methods.

The Performance of Phase II Simple Linear Profile Approaches when Parameters Are Estimated

Previous studies of statistical performance of Phase II simple linear profile approaches were reported only for the case of known profile parameters assumption. The main objective of this article is to evaluate and compare the performance of these approaches when the profile parameters are estimated from an in-control Phase I profile data set. Simulations establish that the performance of these approaches is strongly affected when the parameters are estimated compared to the known parameters case. The in-control performance of the competing approaches significantly deteriorates if estimated parameters are used with control limits intended for known parameters, especially when only a few Phase I samples are used to estimate the parameters. The results show also that some profile monitoring approaches need much larger number of Phase I profiles than other approaches to achieve the expected statistical performance. They also show that the profile monitoring approach proposed by Mahmoud et al. (2010) has generally better out-of-control run length performance than the competing approaches when the estimated parameters are used in the charts design.

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.

Robustness of the EWMA control chart for individual observations

The traditional exponentially weighted moving average (EWMA) chart is one of the most popular control charts used in practice today. The in-control robustness is the key to the proper design and implementation of any control chart, lack of which can render its out-of-control shift detection capability almost meaningless. To this end, Borror et al. [5] studied the performance of the traditional EWMA chart for the mean for i.i.d. data. We use a more extensive simulation study to further investigate the in-control robustness (to non-normality) of the three different EWMA designs studied by Borror et al. [5]. Our study includes a much wider collection of non-normal distributions including light- and heavy-tailed and symmetric and asymmetric bi-modal as well as the contaminated normal, which is particularly useful to study the effects of outliers. Also, we consider two separate cases: (i) when the process mean and standard deviation are both known and (ii) when they are both unknown and estimated from an in-control Phase I sample. In addition, unlike in the study done by Borror et al. [5], the average run-length (ARL) is not used as the sole performance measure in our study, we consider the standard deviation of the run-length (SDRL), the median run-length (MDRL), and the first and the third quartiles as well as the first and the 99th percentiles of the in-control run-length distribution for a better overall assessment of the traditional EWMA chart's in-control performance. Our findings sound a cautionary note to the (over) use of the EWMA chart in practice, at least with some types of non-normal data. A summary and recommendations are provided.

A two-phase method for controlling Erlang-failure processes with high reliability

Monitoring a failure process and measuring its performance are important issues for complex nonrepairable and repairable systems. For a highly reliable process, traditional methods for reliability monitoring and performance measuring become inapplicable. This paper proposes a new two-phase controlling method for monitoring and measuring an Erlang-failure process (EFP). In the first-phase controlling method, a control chart is used to monitor the EFP condition. When special causes of variation have been removed from the EFP and all of the failure times plotted on the control chart lie within the control limits, the EFP is considered to be in control. However, the in-control EFP still likely carries out bad or out-of-lifetime-specification conditions. Thus, its lifetime-specification limit is taken into consideration as the second-phase controlling method for measuring the in-control EFP performance. We propose a lifetime-capability index. Its value has a one-to-one corresponding relationship with the lifetime-conforming rate, which indicates the lifetime performance of this EFP. Without collecting additional data efforts, in-control data gathered from the control chart in the first phase is employed to estimate the lifetime-capability index. To realize main lifetime-capability of the EFP impacting on downstream customers, the lower confidence bound of the estimate of the lifetime-capability index, capturing its minimum lifetime capability, is considered. The advantage of this two-phase method for controlling the failure processes can motivate the manufacturer to develop a reliability-monitoring technique, establish an adequate reliability improvement program and implement an appropriate analysis to ensure its lifetime performance meeting the customers requirement.

A Distribution-Free Phase I Control Chart for Subgroup Scale

Most Phase I control-chart methods are based on the often untested and unreasonable assumption of normally distributed process observations. While there has been some work on distribution-free Phase I procedures for process location, at present, we found no distribution-free Phase I methods for evaluating process scale. We propose a Phase I control chart that is distribution free when the process is in-control. This method can be used to define the in-control state of the process variability and to aid in identifying an in-control reference sample. The proposed scale charting method is compared with the traditional R- and S-charts using Monte Carlo simulation. We show that the in-control performance of the R- and S-charts is poor in Phase I, both when the process data follow a normal distribution and when the distribution deviates from the normal model. Our proposed method achieves desired in-control performance when used with both normal and nonnormal data and is sensitive to subgroup differences in process scale. We offer advice for combining this method with an existing distribution-free Phase I chart for studying process location.

Combined Shewhart–EWMA control charts with estimated parameters

Shewhart and EWMA control charts can be suitably combined to obtain a simple monitoring scheme sensitive to both large and small shifts in the process mean. So far, the performance of the combined Shewhart–EWMA (CSEWMA) has been investigated under the assumption that the process parameters are known. However, parameters are often estimated from reference Phase I samples. Since chart performances may be even largely affected by estimation errors, we study the behaviour of the CSEWMA with estimated parameters in both in- and out-of-control situations. Comparisons with standard Shewhart and EWMA charts are presented. Recommendations are given for Phase I sample size requirements necessary to achieve desired in-control performance.

A phase II nonparametric control chart based on precedence statistics with runs-type signaling rules

Nonparametric control charts do not require knowledge about the shape of the underlying distribution and can thus be attractive in certain situations. Two new Shewhart-type nonparametric control charts are proposed for monitoring the unknown location parameter of a continuous population in Phase II (prospective) applications. The charts are based on control limits given by two specified order statistics from a reference sample, obtained from a Phase I (retrospective) analysis, and using some runs-type signaling rules. The plotting statistic can be any order statistic in a Phase II sample; the median is used here for simplicity and robustness. Exact run length distributions of the proposed charts are derived using conditioning and some results from the theory of runs. Tables are provided for practical implementation of the charts for a given in-control average run length ( ARL 0 ) between 300 and 500. Comparisons of the average run length ARL, the standard deviation of run length (SDRL) and some run length percentiles show that the charts have robust in-control performance and are more efficient when the underlying distribution is t (symmetric with heavier tails than the normal) or gamma (1, 1) (right-skewed). Even for the normal distribution, the new charts are quite competitive. An illustrative numerical example is given. An added advantage of these charts is that they can be applied before all the data are collected which might lead to savings in time and resources in certain applications.

A Note on the Efficiency of Nonparametric Control Chart for Monitoring Process Variability

Nonparametric control chart techniques represent a rather neglected area in Statistical Process Control, in particular with respect to monitoring process variability. There are only very few relevant papers published. The reason for not regarding nonparametric techniques for controlling process variability are investigated in this paper by means of a control chart based on the two sample rank-sum test of Ansari and Bradley. Using the popular 3σ-control limits, the in-control state performance (ARL) is compared by simulation with that of a Shewhart S-chart for different distributions and different sample sizes. Moreover, the efficiency to detect shifts in the variability is investigated. It is shown that the in-control performance is excellent, however, the performance in the out-of-control state is rather poor.

Evaluating and Improving the Unit and Group-Runs Chart

A unit and group-runs (UGR) chart is designed to detect the increase in fraction nonconforming (p) under 100% inspection. This chart significantly outperforms all other attribute-control charts in terms of the zero-state average time to signal (ATSz). Nonetheless, it has been found that the UGR chart would frequently produce a false alarm within a short time period if the process starts from an in-control status, leading to an undesirable in-control behavior. In addition, the out-of-control performance of the UGR chart under the steady-state mode is also unsatisfactory. This article proposes a method to improve the UGR chart using the steady-state ATSs as the objective function. By minimizing ATSs, the in-control performance of the UGR chart, as well as the out-of-control performance under the steady-state mode, could be greatly improved. In the meantime, this UGR chart still performs much more effectively than any other attribute charts if measured by the zero-state ATSz.

A Nonparametric Shewhart-Type Signed-Rank Control Chart Based on Runs

Shewhart-type distribution-free control charts are considered for the known in-control median of a continuous process distribution based on the Wilcoxon signed-rank statistic and some runs type rules. The new charts are more attractive to the practitioner than a basic Shewhart-type signed-rank chart proposed by Bakir (2004), as they offer more desirable (smaller) false alarm rates and (larger) in-control average run-lengths, and can be easily implemented. In addition to being nonparametric, that is with a known and stable in-control performance for all continuous distributions, a simulation study indicates that the proposed charts can have better out-of-control performance than the Shewhart X-bar chart and the basic signed-rank chart for the normal distribution and for some heavy-tailed distributions such as the double exponential and the Cauchy. A numerical example is provided.

Parameter estimation and design considerations in prospective applications of the X¯ chart

The effects of parameter estimation on the in-control performance of the Shewhart X¯ chart are studied in prospective (phase 2 or stage 2) applications via a thorough examination of the attained false alarm rate (AFAR), the conditional false alarm rate (CFAR), the conditional and the unconditional run-length distributions, some run-length characteristics such as the ARL, the conditional ARL (CARL), some selected percentiles including the median, and cumulative run-length probabilities. The examination involves both numerical evaluations and graphical displays. The effects of parameter estimation need to be accounted for in designing the chart. To this end, as an application of the exact formulations, chart constants are provided for a specified in-control average run-length of 370 and 500 for a number of subgroups and subgroup sizes. These will be useful in the implementation of the X¯ chart in practice.

Quantitative Techniques to Evaluate Process Stability

ABSTRACT There have been numerous measures proposed to quantify the capability of a process using statistics such as C p , C pk , and C pm . There are even tests of hypothesis to see if the calculated C p or C pk exceeds some constant. However, there are few quantitative summaries to use as a starting point to assess the overall in-control performance of a process. In this article, we propose three measures that can be used to assess the stability of a process and classify the process parameter as stable or unstable. Test criteria for each of the three measures are suggested, and their overall performance under certain conditions are studied. These concepts are demonstrated using three case studies that are detailed throughout the article.