S. Chakraborti

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.

Phase I Statistical Process Control Charts: An Overview and Some Results

ABSTRACT In practice, Phase I analysis constitutes an integral part of an overall SPC program in which control charts play a crucial role. An overview of the literature on Phase I parametric control charts for univariate variables data is presented. Since the Phase I signaling events are dependent and multiple signaling events are to be dealt with simultaneously in making an in-control or out-of-control decision, the joint distribution of the charting statistics is used to control the false alarm probability, which is defined as the probability of at least one false alarm, while designing the charts. An example is given. Concluding remarks include suggestions regarding future research problems.

Asymptotically Distribution-Free Statistical Inference for Generalized Lorenz Curves

This paper provides asymtotically distribution-free statistical inference procedures for generalized Lorenz curves. Given appropriate measures of income and the income recipient unit are chosen appropriately, the tests allow consensually valid statements regarding social welfare to be made from sample data on the basis of sound inferential procedures. More generally, the results presented here can be applied to test for second degree stochastic dominance. Copyright 1989 by MIT Press.

Run length, average run length and false alarm rate of Shewhart X-bar Chart : Exact derivations by conditioning

The effects of estimation of the control limits on the performance of the popular Shewhart X-bar chart are examined via the average run length and the probability of a false alarm, when one or both of the process mean and variance are unknown. Exact expressions for the run length, the average run length (ARL) and the false alarm rate are obtained, in each case, using expectation by conditioning. Applying Jensens inequality, together with expectation by conditioning, a simple lower bound to the ARL is obtained. This could be useful in designing the charts. The expressions for the exact ARL and the exact probabilities of false alarm are evaluated, using simulations, for various numbers of subgroups and shift sizes. The calculations throw new light on the performance of the Shewhart X-bar chart. Some recommendations are given.

Control Charts for Joint Monitoring of Mean and Variance: An Overview

Abstract In the control chart literature, a number of one-and two-chart schemes has been developed to simultaneously monitor the mean and variance parameters of normally distributed processes. These “joint” monitoring schemes are useful for situations in which special causes can result in a change in both the mean and the variance, and they allow practitioners to avoid the inflated false alarm rate which results from simply using two independent control charts (one each for mean and variance) without adjusting for multiple testing. We present an overview of this literature covering some of the one-and two-chart schemes, including those that are appropriate in parameters known (standards known) and unknown (standards unknown) situations. We also discuss some of the joint monitoring schemes for multivariate processes, autocorrelated data, and individual observations. In addition, noting that normality is often an elusive assumption, we discuss some available nonparametric schemes for jointly monitoring location and scale. We end with a conclusion and some recommendations for areas of further research.

Distribution‐free Phase II CUSUM Control Chart for Joint Monitoring of Location and Scale

A single distribution-free (nonparametric) Shewhart-type chart on the basis of the Lepage statistic is well known in literature for simultaneously monitoring both the location and the scale parameters of a continuous distribution when both of these parameters are unknown. In the present work, we consider a single distribution-free cumulative sum chart, on the basis of the Lepage statistic, referred to as the cumulative sum-Lepage (CL) chart. The proposed chart is distribution-free (nonparametric), and therefore, the in-control properties of the chart remain invariant and known for all continuous distributions. Control limits are tabulated for implementation of the proposed chart in practice. The in-control and out-of-control performance properties of the cumulative sum-Lepage (CL) chart are investigated through simulation studies in terms of the average, the standard deviation, the median, and some percentiles of the run length distribution. Detailed comparison with a competing Shewhart-type chart is presented. Several existing cumulative sum charts are also considered in the performance comparison. The proposed CL chart is found to perform very well in the location-scale models. We also examine the effect of the choice of the reference value (k) on the performance of the CL chart. The proposed chart is illustrated with a real data set. Summary and conclusions are presented. Copyright

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 charts 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.

Design and implementation of CUSUM exceedance control charts for unknown location

Nonparametric control charts provide a robust alternative in practice when the form of the underlying distribution is unknown. Nonparametric CUSUM (NPCUSUM) charts blend the advantages of a CUSUM with that of a nonparametric chart in detecting small to moderate shifts. In this paper, we examine efficient design and implementation of Phase II NPCUSUM charts based on exceedance (EX) statistics, called the NPCUSUM-EX chart. We investigate the choice of the order statistic from the reference (Phase I) sample that defines the exceedance statistic. We see that choices other than the median, such as the 75th percentile, can yield improved performance of the chart in certain situations. Furthermore, observing certain shortcomings of the average run-length, we use the median run-length as the performance metric. The NPCUSUM-EX chart is compared with the NPCUSUM-Rank chart based on the popular Wilcoxon rank-sum statistic. We also study the choice of the reference value, k, of the CUSUM charts. An illustration with real data is provided.

A nonparametric exponentially weighted moving average signed-rank chart for monitoring location

Nonparametric control charts can provide a robust alternative in practice to the data analyst when there is a lack of knowledge about the underlying distribution. A nonparametric exponentially weighted moving average (NPEWMA) control chart combines the advantages of a nonparametric control chart with the better shift detection properties of a traditional EWMA chart. A NPEWMA chart for the median of a symmetric continuous distribution was introduced by Amin and Searcy (1991) using the Wilcoxon signed-rank statistic (see Gibbons and Chakraborti, 2003). This is called the nonparametric exponentially weighted moving average Signed-Rank (NPEWMA-SR) chart. However, important questions remained unanswered regarding the practical implementation as well as the performance of this chart. In this paper we address these issues with a more in-depth study of the two-sided NPEWMA-SR chart. A Markov chain approach is used to compute the run-length distribution and the associated performance characteristics. Detailed guidelines and recommendations for selecting the charts design parameters for practical implementation are provided along with illustrative examples. An extensive simulation study is done on the performance of the chart including a detailed comparison with a number of existing control charts, including the traditional EWMA chart for subgroup averages and some nonparametric charts i.e. runs-rules enhanced Shewhart-type SR charts and the NPEWMA chart based on signs. Results show that the NPEWMA-SR chart performs just as well as and in some cases better than the competitors. A summary and some concluding remarks are given.

A Nonparametric EWMA Sign Chart for Location Based on Individual Measurements

ABSTRACT Nonparametric control charts are useful when there is limited or complete lack of knowledge about the form of the underlying distribution. Though traditional statistical process control (SPC) applications of control charts involve subgrouped data, recent advances have led to more and more instances where individual measurements (data) are collected over time. A two-sided nonparametric exponentially weighted moving average (EWMA) control chart for i.i.d. individual data is proposed based on the sign (SN) statistic. A Markov chain approach is used to determine the run-length distribution of the chart and some associated performance characteristics. An important advantage of the nonparametric EWMA-SN chart is its inherent in-control robustness. In fact, the in-control run-length distribution and hence all of its associated characteristics (e.g., false alarm rate, average, standard deviation, median, etc.) of the chart remain the same for all unknown continuous distributions. In order to aid practical implementation, tables are provided for the charts design parameters. An extensive simulation study shows that on the basis of minimal required assumptions, robustness of the in-control run-length distribution and out-of-control performance, the proposed nonparametric EWMA-SN chart can be a strong contender in many applications where traditional parametric charts are currently used.