Subha Chakraborti

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.

Run Length Distribution and Percentiles: The Shewhart X Chart with Unknown Parameters

ABSTRACT A well-known concern is that important information regarding the performance of a control chart may be missed by focusing too much on the average run length (ARL). This is particularly true since the run length distribution is generally highly right-skewed. The entire run length distribution should be examined for a more complete understanding of the chart performance and this could be facilitated by an examination of a number of representative percentiles including the median, and some functions of the percentiles such as the interquartile range. Khoo (2004) studied the percentiles for the Shewhart chart when the mean and the variance of the process are specified (the so-called “standards known” case). In this paper we closely examine the run length distribution and the percentiles of the Shewhart chart in the case when the process mean and variance are both unknown (the so-called “standards unknown” case) and are therefore estimated. The exact run length c.d.f. is evaluated and plotted for a number of subgroups (m) and a subgroup size (n) of five, for a nominal false alarm rate (FAR) of 0.0027. The run length c.d.f. for the standards known case, which is known to be geometric, is included for comparison. Moreover, a number of specific percentiles are calculated and compared to those in the standards known case. It is seen that for small to moderate values of m, the run length distributions neither dominate nor are dominated by the geometric distribution. When parameters are estimated and the process is in-control, for percentiles of order less than approximately 0.82, the cumulative probability of early runs is larger for small to moderate values of m, whereas for percentiles beyond that, the cumulative probability of late runs is smaller than those under the geometric distribution. In the out-of-control case (for a step shift of 0.5) a similar phenomenon is seen around the percentile of order approximately 0.62. For m = 500 and n = 5, the run length distribution in the standards unknown case converges to that in the standards known case, namely the geometric distribution. An alternate chart design criterion, based on the in-control median run length, is proposed.

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.

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.

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

Abstract 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.

A Distribution-Free Control Chart for the Joint Monitoring of Location and Scale

Traditional statistical process control for variables data often involves the use of a separate mean and a standard deviation chart. Several proposals have been published recently, where a single (combination) chart that is simpler and may have performance advantages, is used. The assumption of normality is crucial for the validity of these charts. In this article, a single distribution-free Shewhart-type chart is proposed for monitoring the location and the scale parameters of a continuous distribution when both of these parameters are unknown. The plotting statistic combines two popular nonparametric test statistics: the Wilcoxon rank sum test for location and the Ansari-Bradley test for scale. Being nonparametric, all in-control properties of the proposed chart remain the same and known for all continuous distributions. Control limits are tabulated for implementation in practice. The in-control and the out-of-control performance properties of the chart are investigated in simulation studies in terms of the mean, the standard deviation, the median, and some percentiles of the run length distribution. The influence of the reference sample size is examined. A numerical example is given for illustration. Summary and conclusions are offered.

Parameter Estimation and Performance of the

Effects of parameter estimation are examined for the well-known p-chart for the fraction nonconforming based on attributes (binary) data. The exact run-length distribution of the chart is obtained for Phase II applications, when the fraction of nonconforming items, p, is unknown, by conditioning on the observed number of nonconformities in a set of reference data (from Phase I) used to estimate p. Numerical illustrations show that the actual performance of the chart can be substantially different from what one would nominally expect, in terms of the false alarm rate and/or the in-control average run-length. Moreover, the performance of the p-chart can be highly degraded in that an exceedingly large number of false alarms are observed, particularly when p is estimated, unless the number of reference observations is substantially large, much larger than what might be commonly used in practice. These results are useful in the study of the reliability of products or systems that involve binary data

p

The double sampling (DS) X@? chart when the process parameters are unknown and have to be estimated from a reference Phase-I dataset is studied. An expression for the run length distribution of the DS X@? chart is derived, by conditioning and taking parameter estimation into account. Since the shape and the skewness of the run length distribution change with the magnitude of the mean shift, the number of Phase-I samples and sample sizes, it is shown that the traditional charts performance measure, i.e. the average run length, is confusing and not a good representation of a typical charts performance. To this end, because the run length distribution is highly right-skewed, especially when the shift is small, it is argued that the median run length (MRL) provides a more intuitive and credible interpretation. From this point of view, a new optimal design procedure for the DS X@? chart with known and estimated parameters is developed to compute the charts optimal parameters for minimizing the out-of-control MRL, given that the values of the in-control MRL and average sample size are fixed. The optimal chart which provides the quickest out-of-control detection speed for a specified shift of interest is designed according to the number of Phase-I samples commonly used in practice. Tables are provided for the optimal chart parameters along with some empirical guidelines for practitioners to construct the optimal DS X@? charts with estimated parameters. The optimal charts with estimated parameters are illustrated with a real application from a manufacturing company.

-Chart for Attributes Data

Studies on the effect of parameter estimation on control-chart performance have mostly focused on the marginal (unconditional) run length (RL) distribution and some associated characteristics. However, once process parameters are estimated from an in-control (IC) reference sample, the RL follows its conditional distribution given the parameter estimates. With this in mind, our focus is the conditional RL distribution of the S and S2 charts. First, we concentrate on the IC conditional RL distribution, which is geometric with parameter (probability of success) αTRUE, the unknown attained false-alarm probability, given the estimate of the process standard deviation. We obtain and examine the distribution of αTRUE as a function of the number and the size of the reference samples. We consider a one-sided prediction interval for αTRUE and, for several sample sizes, obtain the minimum number of reference sample observations that guarantees, with a specified probability, that αTRUE will not exceed the nominal value of the false-alarm probability, αNOM, by more than a prespecified percentage. Next, we argue that (and demonstrate why), in the particular case of S and S2 charts, there is no practical need of a similar analysis of the effects of parameter estimation on their out-of-control performance, and draw some conclusions in terms of guidance for the user.

Optimal design of the double sampling X chart with estimated parameters based on median run length

Nonparametric control charts are considered for the median and other percentiles based on runs of sign statistics above and below the control limits. It is noted that the sign charts are advantageous in certain practical situations. Expressions for the run-length distributions are derived using Markov chain theory; several examples are given. The in-control (IC) and the out-of-control (OOC) performance of these charts are studied and compared to the existing nonparametric Wilcoxon signed-ranked charts of Chakraborti and Eryilmaz (2007) under the normal, the double exponential and the Cauchy distributions, using the average run-length (ARL), the standard deviation of the run-length (SDRL), the false alarm rate (FAR) and some percentiles of the run-length, including the median run-length (MDRL). It is shown that the proposed “runs-rules enhanced” sign charts offer more practically desirable IC ARL (ARL 0) and FAR values and perform better for some heavy-tailed distributions. Some concluding remarks are offered.