Nonparametric control charts based on order statistics: Some advances

ABSTRACT

In this article, we introduce new nonparametric Shewhart-type control charts that take into account the location of two order statistics of the test sample as well as the number of observations in that sample that lie between the control limits. Exact formulae for the alarm rate, the run length distribution and the average run length (ARL) are all derived. A key advantage of the new charts is that, due to its nonparametric nature, the false alarm rate (FAR) and in-control run length distribution is the same for all continuous process distributions. Tables are provided for the implementation of the proposed charts for some typical FAR and ARL values. Furthermore, a numerical study carried out reveals that the new charts are quite flexible and efficient in detecting shifts to Lehmann-type out-of-control situations, while they seem preferable from a robustness point of view in comparison with the distribution-free control chart of Balakrishnan et al. (2009).     

Goodness-of-fit-based outlier detection for Phase I monitoring

ABSTRACT

Nonparametric charts are useful in statistical process control when there is a lack of or limited knowledge about the underlying process distribution. Most existing approaches in the literature of Phase I monitoring assume that outliers have the same distributions as the in-control sample but only differ in location or scale parameters, they may not be effective with distributional changes. This article develops a new procedure based on the integration of the classical Anderson–Darling goodness-of-fit test and the stepwise isolation method. Our proposed procedure is efficient in detecting potential shifts in location, scale, or shape, and thus it offers robust protection against variation in various underlying distributions. The finite sample performance of our method is evaluated through simulations and is compared with that of available outlier detection methods for Phase I monitoring.     

Nonparametric control charts based on runs and Wilcoxon-type rank-sum statistics

In this article, we introduce three new distribution-free Shewhart-type control charts that exploit run and Wilcoxon-type rank-sum statistics to detect possible shifts of a monitored process. Exact formulae for the alarm rate, the run length distribution, and the average run length (ARL) are all derived. A key advantage of these charts is that, due to their nonparametric nature, the false alarm rate (FAR) and in-control run length distribution is the same for all continuous process distributions. Tables are provided for the implementation of the charts for some typical FAR values. Furthermore, a numerical study carried out reveals that the new charts are quite flexible and efficient in detecting shifts to Lehmann-type out-of-control situations.     

A class of distribution-free control charts

Summary.  A class of Shewhart-type distribution-free control charts is considered. A key advantage of these charts is that the in-control run length distribution is the same for all continuous process distributions. Exact expressions for the run length distribution and the average run length (ARL) are derived and properties of the charts are studied via evaluations of the run length distribution probabilities and the ARL. Tables are provided for implementation for some typical ARL values and false alarm rates. The charts proposed are preferable from a robustness point of view, have attractive ARL properties and would be particularly useful in situations where one uses a classical Shewhart   X -chart. A numerical illustration is given.     

A generally weighted moving average exceedance chart

Distribution-free control charts gained momentum in recent years as they are more efficient in detecting a shift when there is a lack of information regarding the underlying process distribution. However, a distribution-free control chart for monitoring the process location often requires information on the in-control process median. This is somewhat challenging because, in practice, any information on the location parameter might not be known in advance and estimation of the parameter is therefore required. In view of this, a time-weighted control chart, labelled as the Generally Weighted Moving Average (GWMA) exceedance (EX) chart (in short GWMA-EX chart), is proposed for detection of a shift in the unknown process location; this chart is based on exceedance statistic when there is no information available on the process distribution. An extensive performance analysis shows that the proposed GWMA-EX control chart is, in many cases, better than its contenders.     

Distribution-free cumulative sum and exponentially weighted moving average control charts based on the Wilcoxon rank-sum statistic using ranked set sampling for monitoring mean shifts

ABSTRACT

Whenever a practitioner is not sure about the underlying process distribution, alternative monitoring schemes that may be used are called nonparametric charts. A nonparametric scheme mostly used to monitor the difference in the means of two samples is called the Wilcoxon rank-sum (WRS). In this paper, we propose nonparametric (or distribution-free) cumulative sum and exponentially weighted moving average charts based on the WRS using ranked set sampling. We thoroughly discuss the performance of the proposed control charts in terms of run-length properties through intensive simulations. Moreover, we conduct an overall performance comparison using the relative mean index and a variety of quality loss functions (for instance, the average extra quadratic loss, average ratio of the average run-length and performance comparison index). The newly proposed charts have very attractive run-length properties and they have better overall performance than their counterparts. An illustrative example is given, as well as an easy-to-use table with optimal design parameters to aid practical implementation.     

The effect of parameter estimation on the performance of one-sided Shewhart control charts for zero-inflated processes

ABSTRACT

Zero-inflated probability models are used to model count data that have an excessive number of zeros. Shewhart-type control charts have been proposed for the monitoring of zero-inflated processes. Usually their performance is evaluated under the assumption of known process parameters. However, in practice, their values are rarely known and they have to be estimated from an in-control historical Phase I sample. In the present paper, we investigate the performance of Shewhart-type control charts for zero-inflated processes with estimated parameters and propose practical guidelines for the statistical design of the examined charts, when the size of the preliminary sample is predetermined.     

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.     

Zone control charts with estimated parameters for detecting prespecified changes in linear profiles

ABSTRACT

In profile monitoring, control charts are proposed to detect unanticipated changes, and it is usually assumed that the in-control parameters are known. However, due to the characteristics of a system or process, the prespecified changes would appear in the process. Moreover, in most applications, the in-control parameters are usually unknown. To overcome these issues, we develop the zone control charts with estimated parameters to detect small shifts of these prespecified changes. The effects of estimation error have been investigated on the performance of the proposed charts. To account for the practitioner-to-practitioner variability, the expected average run length (ARL) and the standard deviation of the average run length (SDARL) is used as the performance metrics. Our results show that the estimation error results in the significant variation in the ARL distribution. Furthermore, in order to adequately reduce the variability, more phase I samples are required in terms of the SDARL metric than that in terms of the expected ARL metric. In addition, more observations on each sampled profile are suggested to improve the charts' performance, especially for small phase I sample sizes. Finally, an illustrative example is given to show the performance of the proposed zone control charts.     

A EWMA control chart based on an auxiliary variable and repetitive sampling for monitoring process location

ABSTRACT

This article develops an exponentially weighted moving average (EWMA) control chart using an auxiliary variable and repetitive sampling for efficient detection of small to moderate shifts in location. A EWMA statistic of a product estimator of the average (which utilities the information of auxiliary variables as well as repetitive sampling) is plotted on the proposed chart. The control chart coefficients of the proposed EWMA chart are determined for two strategic limits known as outer and inner control limits for the target in-control average run length. The performance of the proposed EWMA chart is studied using average run length when a shift occurs in the process average. The efficiency of the developed chart is compared with the competitive existing control charts. The results of the study revealed that proposed EWMA chart is more efficient than others to detect small changes in process mean.     

ARL-unbiased control charts for the monitoring of exponentially distributed characteristics based on type-II censored samples

In this paper, the problem of monitoring process data that can be modelled by exponential distribution is considered when observations are from type-II censoring. Such data are common in many practical inspection environment. An average run length unbiased (ARL-unbiased) control scheme is developed when the in-control scale parameter is known. The performance of the proposed control charts are investigated in terms of the ARL and standard deviation of the run length. The effects of parameter estimation on the proposed control charts are also evaluated. Then, we consider the design of the ARL-unbiased control charts when the in-control scale parameter is estimated. Finally, an example is used to illustrate the implementation of the proposed control charts.     

Run rules based phase II c and np charts when process parameters are unknown

Abstract

The performance of attributes control charts is usually evaluated under the assumption of known process parameters (i.e., the nominal proportion of non conforming units or the nominal average number of nonconformities). However, in practice, these process parameters are rarely known and have to be estimated from an in-control Phase I data set. The major contributions of this paper are (a) the derivation of the run length properties of the Run Rules Phase II c and np charts with estimated parameters, particularly focusing on the ARL, SDRL, and 0.05, 0.5, and 0.95 quantiles of the run length distribution; (b) the investigation of the number m of Phase I samples that is needed by these charts in order to obtain similar in-control ARLs to the known parameters case; and (c) the proposition of new specific chart parameters that allow these charts to have approximately the same in-control ARLs as the ones obtained in the known parameters case.     

Parametric bootstrap process capability index control charts for both mean and dispersion

Abstract

Non-normal processes are common in practice. In this paper, we propose a novel approach to defining bootstrap process capability index (PCI) control charts to monitor the performance of in-control skew normal processes. We use a bootstrap method to calculate phase I control limits of the corresponding PCI control charts. The β-risk curves of the associated PCI control charts will be used to assess the performance of the PCI control charts. We use Monte-Carlo simulation to evaluate the performance of the proposed PCI control charts. A numerical example to illustrate the implementation of the proposed control charts.     

Distribution-free monitoring schemes based on order statistics: a general approach

In this article, we establish a new class of distribution-free Shewhart-type monitoring schemes based on order statistics. The setup of the proposed family of nonparametric control charts is presented in detail. Specific monitoring schemes, already introduced in the literature, are confirmed to be members of the new class. In addition, a new nonparametric monitoring scheme that belongs to the class is established, while explicit formulae for its basic characteristics are reached. The numerical study carried out reveals that the proposed scheme achieves adversarial in-control and out-of-control performance.KEYWORDS: Distribution-free monitoring schemes, average run length, false alarm rate, Lehmann alternatives, statistical process control, order statistics     

Self-starting X-bar control chart based on Six Sigma quality and sometimes pooling procedure

Self-starting control charts have been proposed in the literature to allow process monitoring when only a small amount of relevant data is available. In fact, self-starting charts are useful in monitoring a process quickly, without having to collect a sizable Phase I sample for estimating the in-control process parameters. In this paper, a new self-starting control charting procedure is proposed in which first an effective initial sample is chosen from the perspective of Six Sigma quality, then the successive sample means are either pooled or not pooled (sometimes pooling procedure) for computing next Q-statistics depending upon its signal. It is observed that the sample statistics obtained so from this in-control Phase I situation can serve as more efficient estimators of unknown parameters for Phase II monitoring. An example is considered to illustrate the construction of the proposed chart and to compare its performance with the existing ones.     

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.     

A Robust Control Chart for Monitoring the Mean of an Autocorrelated Process

Residual control charts are frequently used for monitoring autocorrelated processes. In the design of a residual control chart, values of the true process parameters are often estimated from a reference sample of in-control observations by using least squares (LS) estimators. We propose a robust control chart for autocorrelated data by using Modified Maximum Likelihood (MML) estimators in constructing a residual control chart. Average run length (ARL) is simulated for the proposed chart when the underlying process is AR(1). The results show the superiority of the new chart under several situations. Moreover, the chart is robust to plausible deviations from assumed distribution of errors.     

The Effects of Sample Sizes in Phases I and II on p Control Chart Performance

Control charts designed for the properties of non conformities, also called p control charts, are powerful tools used for monitoring a performance of the fraction of non conforming units. Constructing a p chart is often based on the assumption that the in-control proportion of non conforming items (p ₀) is known. In practice, the value of p ₀ is rarely known and is frequently replaced by an estimate from an in-control reference sample in Phase I. This article investigates the effects of sample sizes in both Phase I and Phase II on the performance of p control charts. The conditional and marginal run length distributions are derived and the corresponding numerical studies are conducted. Moreover, the minimal sample sizes required in Phases I and II to ensure adequate statistical performance are proposed when p ₀ = 0.1 and 0.005.     

Formulas for the Design of CUSUM Quality Control Charts

ABSTRACT

In the design of CUSUM control charts, it is common to use charts, tables, or software to find an appropriate critical threshold (h). This article provides an approximate formula to calculate the threshold directly from prespecified values of the reference value (k) and the in-control average run length (ARL₀). Formulas are also provided for choosing k and h from prespecified values of the in-control and out-of-control average run lengths.     

Progressive Variance Control Charts for Monitoring Process Dispersion

In a process, the deviation from location or scale parameters affects the quality of the process and waste resources. So it is essential to monitor such processes for possible changes due to any assignable causes. Control charts are the most famous tool used to meet this intention. It is useless to monitor process location until the assurance that process dispersion is in-control. This study proposes some new two-sided memory control charts named as progressive variance (PV) control charts which are based on sample variance to monitor changes in process dispersion assuming normality of quality characteristic to be monitored. Simulation studies are made, and an example is discussed to evaluate the performance of the proposed charts. The comparison of the proposed chart is made with exponentially weighted moving average- and cumulative sum-type charts for process dispersion. The study shows that performance of the proposed charts are uniformly better than its competitors for detecting positive shifts while for detecting negative shift in the variance their performance is better for small shifts and reasonably good for moderated shifts.