Robust algorithms for economic designing of a nonparametric control chart for abrupt shift in location

The existing statistical process control procedures typically rely on the fundamental assumption of a parametric distribution of the quality characteristic. However, when there is a lack of knowledge about the underlying distribution (as full knowledge is not available in practice), the performance of these parametric charts is very likely to be heavily degraded. Motivated by this problem, a one-sided nonparametric monitoring procedure using the single sample sign statistic is proposed for detecting a shift in the location parameter of a continuous distribution. An economic model of the control chart is developed to optimize the sample size, sampling interval, and control limits. Three data-dependent estimation approaches for the unknown parameter are evaluated and discussed. Simulation results exhibit that our proposed procedure generally performs well under a great variety of continuous distributions and hence it is recommended as an alternative scheme especially when the knowledge of the underlying distribution is imperfect. Furthermore, beneficial recommendations of estimation approach selection are provided for practical implementation of the control chart.     

Multivariate Control Charts Based on Hybrid Novelty Scores

We propose a new nonparametric multivariate control chart that integrates a novelty score. The proposed control chart uses as its monitoring statistic a hybrid novelty score, calculated based on the distance to local observations as well as on the distance to the convex hull constructed by its neighbors. The control limits of the proposed control chart were established based on a bootstrap method. A rigorous simulation study was conducted to examine the properties of the proposed control chart under various scenarios and compare it with existing multivariate control charts in terms of average run length (ARL) performance. The simulation results showed that the proposed control chart outperformed both the parametric and nonparametric Hotelling's T ² control charts, especially in nonnormal situations. Moreover, experimental results with real semiconductor data demonstrated the applicability and effectiveness of the proposed control chart. To increase the capability to detect small mean shift, we propose an exponentially weighted hybrid novelty score control chart. Simulation results indicated that exponentially weighted hybrid score charts outperformed the hybrid novelty score based control charts.     

Parametric control charts

Standard control charts are based on the assumption that the observations are normally distributed. In practice, normality often fails and consequently the false alarm rate is seriously in error. Application of a nonparametric approach is only possible with many Phase I observations. Since nowadays such very large sample sizes are usually not available, there is need for an intermediate approach by considering a larger parametric model containing the normal family as a submodel. In this paper control limits are presented in such a larger parametric model, the so called normal power family. Correction terms are derived, taking into account that the parameters are estimated. Simulation results show that the control limits are accurate, not only in the considered parametric family, but also for common distributions outside the parametric family, thus covering a broad class of distributions.     

A control chart for variance based on squared ranks

A nonparametric control chart for variance is proposed. The chart is constructed following the change-point approach through the recursive use of the squared ranks test for variance. It is capable of detecting changes in the behaviour of individual observations with performance similar to a self-starting CUSUM chart for scale when normality is assumed, and a relatively better power when assessing nonnormal observations. A comparison is also made with two equivalent nonparametric charts based on Mood and Ansari-Bradley statistics. When dealing with symmetrical distributions, the proposed chart shows smaller (better) out-of-control average run length (ARL), and a competing performance otherwise. In addition, sensitivity to changes in mean and variance at the same time was tested. Extensive Monte Carlo simulation was used to measure performance, and a practical example is provided to illustrate how the proposed control chart can be implemented in practice.     

A new approach for monitoring process variance

ABSTRACT

Control charts are effective tools for signal detection in both manufacturing processes and service processes. Much service data come from a process with variables having non-normal or unknown distributions. The commonly used Shewhart variable control charts, which depend heavily on the normality assumption, should not be properly used in such circumstances. In this paper, we propose a new variance chart based on a simple statistic to monitor process variance shifts. We explore the sampling properties of the new monitoring statistic and calculate the average run lengths (ARLs) of the proposed variance chart. Furthermore, an arcsine transformed exponentially weighted moving average (EWMA) chart is proposed because the ARLs of this modified chart are more intuitive and reasonable than those of the variance chart. We compare the out-of-control variance detection performance of the proposed variance chart with that of the non-parametric Mood variance (NP-M) chart with runs rules, developed by Zombade and Ghute [Nonparametric control chart for variability using runs rules. Experiment. 2014;24(4):1683–1691], and the nonparametric likelihood ratio-based distribution-free exponential weighted moving average (NLE) chart and the combination of traditional exponential weighted moving average (EWMA) mean and EWMA variance (CEW) control chart proposed by Zou and Tsung [Likelihood ratio-based distribution-free EWMA control charts. J Qual Technol. 2010;42(2):174–196] by considering cases in which the critical quality characteristic has a normal, a double exponential or a uniform distribution. Comparison results showed that the proposed chart performs better than the NP-M with runs rules, and the NLE and CEW control charts. A numerical example of service times with a right-skewed distribution from a service system of a bank branch in Taiwan is used to illustrate the application of the proposed variance chart and of the arcsine transformed EWMA chart and to compare them with three existing variance (or standard deviation) charts. The proposed charts show better detection performance than those three existing variance charts in monitoring and detecting shifts in the process variance.     

Distribution-Free Exceedance CUSUM Control Charts for Location

Distribution-free (nonparametric) control charts can be useful to the quality practitioner when the underlying distribution is not known. A Phase II nonparametric cumulative sum (CUSUM) chart based on the exceedance statistics, called the exceedance CUSUM chart, is proposed here for detecting a shift in the unknown location parameter of a continuous distribution. The exceedance statistics can be more efficient than rank-based methods when the underlying distribution is heavy-tailed and/or right-skewed, which may be the case in some applications, particularly with certain lifetime data. Moreover, exceedance statistics can save testing time and resources as they can be applied as soon as a certain order statistic of the reference sample is available. Guidelines and recommendations are provided for the chart's design parameters along with an illustrative example. The in- and out-of-control performances of the chart are studied through extensive simulations on the basis of the average run-length (ARL), the standard deviation of run-length (SDRL), the median run-length (MDRL), and some percentiles of run-length. Further, a comparison with a number of existing control charts, including the parametric CUSUM chart and a recent nonparametric CUSUM chart based on the Wilcoxon rank-sum statistic, called the rank-sum CUSUM chart, is made. It is seen that the exceedance CUSUM chart performs well in many cases and thus can be a useful alternative chart in practice. A summary and some concluding remarks are given.     

Properties of the rank transformation in factorial analysis of covariance

Real world data often fail to meet the underlying assumption of population normality. The Rank Transformation (RT) procedure has been recommended as an alternative to the parametric factorial analysis of covariance (ANCOVA). The purpose of this study was to compare the Type I error and power properties of the RT ANCOVA to the parametric procedure in the context of a completely randomized balanced 3 × 4 factorial layout with one covariate. This study was concerned with tests of homogeneity of regression coefficients and interaction under conditional (non)normality. Both procedures displayed erratic Type I error rates for the test of homogeneity of regression coefficients under conditional nonnormality. With all parametric assumptions valid, the simulation results demonstrated that the RT ANCOVA failed as a test for either homogeneity of regression coefficients or interaction due to severe Type I error inflation. The error inflation was most severe when departures from conditional normality were extreme. Also associated with the RT procedure was a loss of power. It is recommended that the RT procedure not be used as an alternative to factorial ANCOVA despite its encouragement from SAS, IMSL, and other respected sources.     

Nonparametric quality control charts based on the sign statistic

Nonparametric control chart are presented for the problem of detecting changes in the process median (or mean), or changes in the process variability when samples are taken at regular time intervals. The proposed procedures are based on sign-test statistics computed for each sample, and are used in Shewhart and cumulative sum control charts. When the process is in control the run length distributions for the proposed nonparametric control charts do not depend on the distribution of the observations. An additional advantage of the non-parametric control charts is that the variance of the process does not need to be established in order to set up a control chart for the mean. Comparisons with the corresponding parametric control charts are presented. It is also shown that curtailed sampling plans can considerably reduce the expected number of observations used in the Shewhart control schemes based on the sign statistic.     

A Semi‐empirical Bayesian Chart to Monitor Weibull Percentiles

This paper develops a Bayesian control chart for the percentiles of the Weibull distribution, when both its in‐control and out‐of‐control parameters are unknown. The Bayesian approach enhances parameter estimates for small sample sizes that occur when monitoring rare events such as in high‐reliability applications. The chart monitors the parameters of the Weibull distribution directly, instead of transforming the data as most Weibull‐based charts do in order to meet normality assumption. The chart uses accumulated knowledge resulting from the likelihood of the current sample combined with the information given by both the initial prior knowledge and all the past samples. The chart is adapting because its control limits change (e.g. narrow) during Phase I. An example is presented and good average run length properties are demonstrated.     

Restricted Estimation in Partially Linear Errors-in-Variables Models

As a compromise between parametric regression and nonparametric regression, partially linear models are frequently used in statistical modelling. This article considers statistical inference for this semiparametric model when the linear covariate is measured with additive error and some additional linear restrictions on the parametric component are assumed to hold. We propose a restricted corrected profile least-squares estimator for the parametric component, and study the asymptotic normality of the estimator. To test hypothesis on the parametric component, we construct a Wald test statistic and obtain its limiting distribution. Some simulation studies are conducted to illustrate our approaches.     

A non parametric CUSUM control chart based on the Mann–Whitney statistic

We consider a novel univariate non parametric cumulative sum (CUSUM) control chart for detecting the small shifts in the mean of a process, where the nominal value of the mean is unknown but some historical data are available. This chart is established based on the Mann–Whitney statistic as well as the change-point model, where any assumption for the underlying distribution of the process is not required. The performance comparisons based on simulations show that the proposed control chart is slightly more effective than some other related non parametric control charts.     

A bootstrap control chart for inverse Gaussian percentiles

The conventional Shewhart-type control chart is developed essentially on the central limit theorem. Thus, the Shewhart-type control chart performs particularly well when the observed process data come from a near-normal distribution. On the other hand, when the underlying distribution is unknown or non-normal, the sampling distribution of a parameter estimator may not be available theoretically. In this case, the Shewhart-type charts are not available. Thus, in this paper, we propose a parametric bootstrap control chart for monitoring percentiles when process measurements have an inverse Gaussian distribution. Through extensive Monte Carlo simulations, we investigate the behaviour and performance of the proposed bootstrap percentile charts. The average run lengths of the proposed percentage charts are investigated.     

An Examination of the Robustness to Non Normality of the EWMA Control Charts for the Dispersion

ABSTRACT

The EWMA control chart is used to detect small shifts in a process. It has been shown that, for certain values of the smoothing parameter, the EWMA chart for the mean is robust to non normality. In this article, we examine the case of non normality in the EWMA charts for the dispersion. It is shown that we can have an EWMA chart for dispersion robust to non normality when non normality is not extreme.     

An Improved Distribution-free EWMA Mean Chart

Control charts are effective tools for signal detection in both manufacturing processes and service processes. Much service data come from a process with variables having nonnormal or unknown distributions. The commonly used Shewhart variable control charts, which depend heavily on the normality assumption, should not be properly used here. In this article, we propose an improved asymmetric EWMA mean chart based on a simple statistic to monitor process mean shift. We explored the sampling properties of the new monitoring statistic and calculated the average run lengths of the proposed asymmetric EWMA mean chart. We recommend the proposed improved asymmetric EWMA mean chart because the average run lengths of the modified charts are more accurate and reasonable than those of the five existed mean charts. A numerical example of service times with a right skewed distribution from a service system of a bank branch is used to illustrate the application of the improved asymmetric EWMA mean chart and to compare it with the five existing mean charts. The proposed chart showed better detection performance than those of the five existing mean charts in monitoring and detecting shifts in the process mean.     

Non parametric double generally weighted moving average sign charts based on process proportion

Non parametric control charts have received increasing attention in the field of statistical process control. This paper presents a non parametric double generally weighted moving average (DGWMA) sign chart for monitoring small deviations when the quality characteristics of a process are unknown. The statistical performance of the non parametric DGWMA sign chart is evaluated and compared with those of other charts, including the exponentially weighted moving average (EWMA), generally weighted moving average (GWMA), and double EWMA (DEWMA) sign charts. Simulation studies indicate that the non parametric DGWMA sign chart with a large design and median adjustment parameters is always more sensitive than other charts in detecting small changes.     

Minimum control charts

Standard control charts are very sensitive to estimation effects and/or deviations from normality. Hence a program has been carried out to remedy these problems. This is quite adequate in most circumstances, but not in all. In the present paper, the remaining complication is attacked: what to do if a nonparametric approach is indicated, but too few control observations are available for the estimation step? It is shown that grouping the observations during the monitoring phase works well. Surprisingly, rather than using the group averages, it is definitely preferable to compare the minimum for each group to a suitably chosen upper control limit. (And in the two-sided case, also the maximum to an analogous lower control limit.) This ‘minimum control chart’ is demonstrated to be quite attractive: it is easy to explain and to implement. Moreover, while it is truly nonparametric, its power of detection is comparable to that of the customary, normality assuming, charts based on averages.     

A new nonparametric synthetic EWMA control chart for monitoring process mean

The exponentially weighted moving average (EWMA) control charts are widely used in chemical and process industries because of their excellent speed in catching small to moderate shifts in the process target. In usual practice, many data come from a process where the monitoring statistic is non-normally distributed or it follows an unknown probability distribution. This necessitates the use of distribution-free/nonparametric control charts for monitoring the deviations from the process target. In this paper, we integrate the existing EWMA sign chart with the conforming run length chart to propose a new synthetic EWMA (SynEWMA) sign chart for monitoring the process mean. The SynEWMA sign chart encompasses the synthetic sign and EWMA sign charts. Monte Carlo simulations are used to compute the run length profiles of the SynEWMA sign chart. Based on a comprehensive comparison, it turns out that the SynEWMA sign chart is able to perform substantially better than the existing EWMA sign chart. Both real and simulated data sets are used to explain the working and implementation of existing and proposed control charts.     

Empirical Non-Parametric Control Charts: Estimation Effects and Corrections

Owing to the extreme quantiles involved, standard control charts are very sensitive to the effects of parameter estimation and non-normality. More general parametric charts have been devised to deal with the latter complication and corrections have been derived to compensate for the estimation step, both under normal and parametric models. The resulting procedures offer a satisfactory solution over a broad range of underlying distributions. However, situations do occur where even such a large model is inadequate and nothing remains but to consider non- parametric charts. In principle, these form ideal solutions, but the problem is that huge sample sizes are required for the estimation step. Otherwise the resulting stochastic error is so large that the chart is very unstable, a disadvantage that seems to outweigh the advantage of avoiding the model error from the parametric case. Here we analyse under what conditions non-parametric charts actually become feasible alternatives for their parametric counterparts. In particular, corrected versions are suggested for which a possible change point is reached at sample sizes that are markedly less huge (but still larger than the customary range). These corrections serve to control the behaviour during in-control (markedly wrong outcomes of the estimates only occur sufficiently rarely). The price for this protection will clearly be some loss of detection power during out-of-control. A change point comes in view as soon as this loss can be made sufficiently small.     

An adaptive exponentially weighted moving average control chart for monitoring process variances

The exponentially weighted moving average (EWMA) control chart is efficient in detecting small changes in process parameters but less efficient when the changes are relatively large, due to what is known as the inertia problem. To diminish the inertia, an adaptive EWMA (AEWMA) chart has been proposed for monitoring process locations to improve over the traditional EWMA charts. The basic idea of the AEWMA scheme is to dynamically weight the past observations according to a suitable function of the current prediction error. This article extends the idea of the AEWMA chart for monitoring process locations to the case of monitoring process dispersion. A Markov chain model is established to analyze and design the suggested chart. It is shown that the AEWMA dispersion chart performs better than the EWMA and other dispersion charts in terms of its ability to perform relatively well at both small and large changes in process dispersion.     

Bootstrap-Based T 2 Multivariate Control Charts

Control charts have been used effectively for years to monitor processes and detect abnormal behaviors. However, most control charts require a specific distribution to establish their control limits. The bootstrap method is a nonparametric technique that does not rely on the assumption of a parametric distribution of the observed data. Although the bootstrap technique has been used to develop univariate control charts to monitor a single process, no effort has been made to integrate the effectiveness of the bootstrap technique with multivariate control charts. In the present study, we propose a bootstrap-based multivariate T ² control chart that can efficiently monitor a process when the distribution of observed data is nonnormal or unknown. A simulation study was conducted to evaluate the performance of the proposed control chart and compare it with a traditional Hotelling's T ² control chart and the kernel density estimation (KDE)-based T ² control chart. The results showed that the proposed chart performed better than the traditional T ² control chart and performed comparably with the KDE-based T ² control chart. Furthermore, we present a case study to demonstrate the applicability of the proposed control chart to real situations.