Distribution-free Control Charts Research Articles

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.

Distribution-Free Multivariate Control Chart Based on Change-Point Model

It is common to monitor several quality characteristics of a process simultaneously in modern quality control, and it is called multivariate statistical process control (MSPC) in the literature. A change-point control chart for detecting shifts in the mean of a multivariate statistical process is developed for the case where the nominal value of the mean is unknown but some historical samples are available. This control chart is called distribution-free multivariate control chart based on change-point model. Its distribution robustness is a significant advantage where we usually know nothing about the underlying distribution. And the simulated results show that this approach has a good performance across the range of possible shifts.

Diagnosing Problems of Distribution-Free Multivariate Control Chart

In order to solve the problem of only have a few historical data that can be used in multivariate process monitoring, a new distribution-free multivariate control chart has been proposed. And in the control chart structure the control limits are determined on-line with the future observations and the historical data. Therefore, the proposed control chart has very important application in practice. However, the research doesn’t study the problem of the fault diagnosis after the control chart alarms. So we use LASSO-based diagnostic framework to identify when a detected shift has occurred and to isolate the shifted components.

Likelihood ratio-based distribution-free sequential change-point detection

Most existing control charts are for monitoring location or scale parameters, rather than any change in process distribution such as shift in shape. Goodness-of-fit (GOF) test can detect any change in distribution. This paper develops a new distribution-free control chart by integrating a powerful two-sample nonparametric likelihood ratio GOF test into the effective change-point model. Our proposed chart is easy in computation, convenient to use, and very efficient in detecting any change in process distribution, including shifts in location, scale, and shape. It is also robust in detecting various magnitudes of shifts and especially powerful in monitoring any distributional change involving a decrease in scale.

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.

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.

Distribution-free monitoring of univariate processes

We consider statistical process control (SPC) of univariate processes when observed data are not normally distributed. Most existing SPC procedures are based on the normality assumption. In the literature, it has been demonstrated that their performance is unreliable in cases when they are used for monitoring non-normal processes. To overcome this limitation, we propose two SPC control charts for applications when the process data are not normal, and compare them with the traditional CUSUM chart and two recent distribution-free control charts. Some empirical guidelines are provided for practitioners to choose a proper control chart for a specific application with non-normal data.

A Nonparametric Control Chart for Location Based on Sub-Samples of Size Two

Control charts are widely used in statistical process control to detect changes in a production process and for monitoring a process to make sure that it is in control. In conventional statistical process control, the pattern of chance causes is often assumed to follow normal distribution. It is well known that the assumption of normality or any specific parametric form for the process distribution is too restrictive. In such situations, distribution-free or nonparametric control charts can serve the purpose better. In this paper, distribution-free control charts are developed based on a class of one-sample test statistics with sub-samples of size two due to Mehra, Prasad and Madhava Rao (Austral. J. Statist. 32: 373392, 1990). The control charts based on their statistic (-chart) are easy to understand and to use. The performance of the proposed procedures is studied through the average run length, which is the expected number of samples required by the procedure to signal out of control. It is observed that the performance of the proposed chart is better than the existing charts in the literature.

Likelihood Ratio-Based Distribution-Free EWMA Control Charts

Nonparametric or distribution-free 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 are for monitoring location parameters. They may not be effective with a change of distribution over time in many applications. This paper develops a new distribution-free control chart based on the integration of a powerful nonparametric goodness-of-fit test and the exponentially weighted moving-average (EWMA) control scheme. Benefiting from certain desirable properties of the test and the proposed charting statistic, our proposed control chart is computationally fast, convenient to use, and efficient in detecting potential shifts in location, scale, and shape. Thus, it offers robust protection against variation in various underlying distributions. Numerical studies and a real-data example show that the proposed approaches are quite effective in industrial applications, particularly in start-up and short-run situations.

Distribution-Free Phase I Control Charts for Subgroup Location

Much of the work in statistical quality control is dependent on the proper completion of a Phase I study. Many Phase I control charts are based on an implicit assumption of normally distributed process observations. In the beginning stages of process control, little information is available about the process and the normality assumption may not be reasonable. Existing robust and distribution-free control charts are concerned with the establishment of Phase II control limits that are robust to nonnormality or outliers from the Phase I sample. Our literature review revealed no purely distribution-free Phase I control-chart methods. We propose a distribution-free method for defining the in-control state of a process and identifying an in-control reference sample. The resultant reference sample can be used to estimate the process parameters for the Phase II procedure of choice. The proposed rank-based method is compared with the traditional X chart using Monte Carlo simulation. The rank-based method compares favorably to the X chart when the process is normally distributed and performs better than the X chart in many situations when the process distribution is skewed or heavy tailed.

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.

Distribution-Free Quality Control Charts Based on Signed-Rank-Like Statistics

In this article we propose three distribution-free (or nonparametric) statistical quality control charts for monitoring a process center when an in-control target center is not specified. These charts are of the Shewhart-type, the exponentially moving average-type, and the cumulative sum-type. The constructions of the proposed charts require the availability of an initial reference sample taken when the process was operating in-control to calculate an estimator for the unknown in-control target process center. This estimated center is then used in the calculation of signed-rank-like statistics based on grouped observations taken periodically from the process output. As long as the in-control process underlying distribution is continuous and symmetric, the proposed charts have a constant in-control average run length and a constant false alarm rate irrespective of the process underlying distribution. Other advantages of the proposed distribution-free charts include their robustness against outliers and their superior efficiency over the traditional normal-based control charts when applied to processes with moderate- or heavy-tailed underlying distributions, such as the double exponential or the Cauchy distributions.

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 Distribution-Free Shewhart Quality Control Chart Based on Signed-Ranks

Since their inception by Walter Shewhart in the late 1920s, most control chart developments have been distribution-based procedures in the sense that the process output is assumed to follow a specified probability distribution (normal for continuous measurements and binomial or Poisson for attribute data). Due to Deming's influence and their widespread adoption as one of the seven basic tools of total quality management (TQM), control charts have been applied to processes where data may be markedly nonnormal. In this article, we propose a distribution-free (or nonparametric) statistical quality control chart for monitoring a process center. The proposed chart is of the Shewhart type and is based on the signed-ranks of grouped observations. The exact false alarm rate and the in-control average run length of the proposed chart are computed by using the null distribution of the well-known Wilcoxon signed-rank statistic. The out-of-control run lengths are computed exactly for normal underlying distributions and by simulation for uniform, double exponential, and Cauchy shift alternatives. Efficiency studies show that the proposed chart is more efficient than the traditional Shewhart X-bar chart under heavy-tailed distributions (the double exponential and the Cauchy) but is less efficient under light-tailed distributions (the uniform and the normal).

Nonparametric Control Charts: An Overview and Some Results

We present an overview of the literature on nonparametric or distribution-free control charts for univariate variables data. We highlight various advantages of these charts while pointing out some of the disadvantages of the more traditional, distribution-based control charts. Specific observations are made in the course of review of articles and constructive criticism is offered so that opportunities for further research can be identified. Connections to some areas of active research are made, such as sequential analysis, that are relevant to process control. We hope that this article leads to a wider acceptance of distribution-free control charts among practitioners and serves as an impetus to future research and development in this area.