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.     

Modified S-charts for controlling process variability

To help in the detection of variance increases and decreases, three modified versions of traditional Shewhart S-charts are evaluated in terms of their average run length values. One scheme uses control limits based on equal tail chi-square distribution probabilities. The second uses control limits based on unequal tail probabilities. The third uses warning limits based on equal tail probabilities, but requires two successive points beyond the warning limit to give an out-of-control signal. They all result in better average run length values than the traditional S-chart. Also, if the only concern is the detection of variance increases, then both S-charts and warning limit charts without lower control limits are shown to have better average run length values than those of the traditional charts.     

EWNA charts for monitoring the mean and the autocovariances of stationary processes

In this paper various types of EWMA control charts are introduced for the simultaneous monitoring of the mean and the autocovariances. The target process is assumed to be a stationary process up to fourth-order or an ARMA process with heavy tailed innovations. The case of a Gaussian process is included in our results as well. The charts are compared within a simulation study. As a measure of the performance the average run length is taken. The target process is an ARMA (1,1) process with Student-t distributed innovations. The behavior of the charts is analyzed with respect to several out-of-control models. The best design parameters are determined for each chart. Our comparisons show that the multivariate EWMA chart applied to the residuals has the best overall performance.     

Evaluation of control charts under linear trend

The performance of several control charting schemes is studied when the process mean changes as a linear trend. The control charts considered include the Shewhart chart, the Shewhart chart supplemented with runs rules, the cumulative sum (CUSUM) chart, the exponentially weighted moving average (EWMA) chart, and a generalized control chart.     

Shewhart x-charts with estimated process variance

Properties of the Shewhart X-chart for controlling the mean of a process with a normal distribution are investigated for the situation where the process variance Ó²must be estimated from initial sample data. The control limits of the X-chart depend on the estimate of Ó²and thus, unlike the case when Ó²is known, the X-chart is not equivalent to a sequence of independent tests. When Ó²is estimated the distribution of the run length is not geometric and cannot be characterized simply in terms of the probability of a signal at a given point. The average run length (ARL) for the X-chart is expressed in terms of an integral involving the normal cdf, and it is shown that the chart signals with

probability one, but the ARL may not be finite if the size of the 2 sample used to estimate Ó²is sufficiently small. In addition, certain bounds for the ARL are also derived. Numerical integration is use to show that the effect of using small sample sizes in estimating Ó²is to increase the ARL and the variance of the run length distribution     

The performance of multivariate CUSUM control charts with estimated parameters

In this paper, we study the effect of estimating the vector of means and the variance–covariance matrix on the performance of two of the most widely used multivariate cumulative sum (CUSUM) control charts, the MCUSUM chart proposed by Crosier [Multivariate generalizations of cumulative sum quality-control schemes, Technometrics 30 (1988), pp. 291–303] and the MC1 chart proposed by Pignatiello and Runger [Comparisons of multivariate CUSUM charts, J. Qual. Technol. 22 (1990), pp. 173–186]. Using simulation, we investigate and compare the in-control and out-of-control performances of the competing charts in terms of the average run length measure. The in-control and out-of-control performances of the competing charts deteriorate significantly if the estimated parameters are used with control limits intended for known parameters, especially when only a few Phase I samples are used to estimate the parameters. We recommend the use of the MC1 chart over that of the MCUSUM chart if the parameters are estimated from a small number of Phase I samples.     

The optimal choice of negative binomial charts for monitoring high-quality processes

Good control charts for high quality processes are often based on the number of successes between failures. Geometric charts are simplest in this respect, but slow in recognizing moderately increased failure rates p. Improvement can be achieved by waiting until r>1 failures have occurred, i.e. by using negative binomial charts. In this paper we analyze such charts in some detail. On the basis of a fair comparison, we demonstrate how the optimal r is related to the degree of increase of p. As in practice p will usually be unknown, we also analyze the estimated version of the charts. In particular, simple corrections are derived to control the nonnegligible effects of this estimation step.     

On control charts for monitoring the variance of a time series

In this paper we derive control charts for the variance of a Gaussian process using the likelihood ratio approach, the generalized likelihood ratio approach, the sequential probability ratio method and a generalized sequential probability ratio procedure, the Shiryaev–Roberts procedure and a generalized modified Shiryaev–Roberts approach. Recursive presentations for the calculation of the control statistics are given for autoregressive processes of order 1. In an extensive simulation study these schemes are compared with existing control charts for the variance. In order to asses the performance of the schemes both the average run length and the average delay are used.     

A study of economic design of control charts for cumulative count of conforming items

In this paper the economic design of Cumulative Count of Conforming (CCC) control charts to maintain the current control of fraction nonconforming of a process is studied. CCC chart is an attribute chart for monitoring high quality processes by plotting the cumulative count of conforming items between two nonconforming ones on a suitable chart. A process model is proposed to obtain an appropriate loss function. An alogorithm to search for the optimal setting of the sampling and control parameters is derived. Numerical illustrations of the method and some properties of the optimal economic design are provided.     

On the improvement of one-sided S control charts

The most common charting procedure used for monitoring the variance of the distribution of a quality characteristic is the S control chart. As a Shewhart-type control chart, it is relatively insensitive in the quick detection of small and moderate shifts in process variance. The performance of the S chart can be improved by supplementing it with runs rules or by varying the sample size and the sampling interval. In this work, we introduce and study one-sided adaptive S control charts, supplemented or not with one powerful runs rule, for detecting increases or decreases in process variation. The properties of the proposed control schemes are obtained by using a Markov chain approach. Furthermore, a practical guidance for the choice of the most suitable control scheme is also provided.     

Moving range EWMA control charts for monitoring the Weibull shape parameter

In this article, we propose an exponentially weighted moving average (EWMA) control chart for the shape parameter β of Weibull processes. The chart is based on a moving range when a single measurement is taken per sampling period. We consider both one-sided (lower-sided and upper-sided) and two-sided control charts. We perform simulations to estimate control limits that achieve a specified average run length (ARL) when the process is in control. The control limits we derive are ARL unbiased in that they result in ARL that is shorter than the stable-process ARL when β has shifted. We also perform simulations to determine Phase I sample size requirements if control limits are based on an estimate of β. We compare the ARL performance of the proposed chart to that of the moving range chart proposed in the literature.     

Performance of the zone control chart

Average run lengths of the zone control chart are presented, The performance of this chart is compared with that of several Shewhart charts with and without runs rules, It is shown that the standard zone control chart has performance similar to some even simpler charts and a much higher false alarm rate than the Shewhart chart with all of the common runs rules. It is also shown that a slightly modified zone control chart outperforms the Shewhart chart with the common runs rules.     

Empirical nonparametric control charts for high-quality processes

For attribute data with (very) small failure rates often control charts are used which decide whether to stop or to continue each time r failures have occurred, for some r?1. Because of the small probabilities involved, such charts are very sensitive to estimation effects. This is true in particular if the underlying failure rate varies and hence the distributions involved are not geometric. Such a situation calls for a nonparametric approach, but this may require far more Phase I observations than are typically available in practice. In the present paper it is shown how this obstacle can be effectively overcome by looking not at the sum but rather at the maximum of each group of size r.     

Transition probabilities and ARL performance of Six Sigma zone control charts

In this article, Six Sigma zone control charts (SSZCCs) are proposed for world class organizations. The transition probabilities are obtained using the Markov chain approach. The Average Run Length (ARL) values are then presented. The ARL performance of the proposed SSZCCs and the standard Six Sigma control chart (SSCC) without zones or run rules is studied. The ARL performance of these charts is then compared with those of the other standard zone control charts (ZCCs), the modified ZCC and the traditional Shewhart control chart (SCC) with common run rules. As expected, it is shown that the proposed SSZCC outperforms the standard SSCC without zones or run rules for process shifts of any magnitude. When compared to the other standard ZCCs and the Shewhart chart with common run rules, it is observed that the proposed SSZCCs have much higher false alarm rates for smaller shifts and hence they prevent unwanted process disturbances. The application of the proposed SSZCC is illustrated using a real time example.     

One-sided EWMA control charts for monitoring Poisson processes with varying sample sizes

ABSTRACT

Recently considerable research has been devoted to monitoring increases of incidence rate of adverse rare events. This paper extends some one-sided upper exponentially weighted moving average (EWMA) control charts from monitoring normal means to monitoring Poisson rate when sample sizes are varying over time. The approximated average run length bounds are derived for these EWMA-type charts and compared with the EWMA chart previously studied. Extensive simulations have been conducted to compare the performance of these EWMA-type charts. An illustrative example is given.     

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.     

Runs tests for group control charts

When a process consists of several identical streams that are not highly correlated, an alternative to using separate control charts for each stream is to use a group control chart. Rather than plotting sample means from each stream at any time point, one could plot only the largest and/or smallest sample mean from among all the streams. Using the theory of stochastic processes and majorization together with numerical methods, the properties of a test that signals if r consecutive extreme values come from the same stream are examined. Both one and two-sided cases are considered. Average run lengths (ARL's), the least favorable configuration of the stream (population) means, and sample sizes necessary to have specified in-control and out-of-control ARL's are obtained. A test that signals if r-1 out of r consecutive extreme values come from the same stream is also considered     

The effect of Phase I sample size on the run length performance of control charts for autocorrelated data

Traditional control charts assume independence of observations obtained from the monitored process. However, if the observations are autocorrelated, these charts often do not perform as intended by the design requirements. Recently, several control charts have been proposed to deal with autocorrelated observations. The residual chart, modified Shewhart chart, EWMAST chart, and ARMA chart are such charts widely used for monitoring the occurrence of assignable causes in a process when the process exhibits inherent autocorrelation. Besides autocorrelation, one other issue is the unknown values of true process parameters to be used in the control chart design, which are often estimated from a reference sample of in-control observations. Performances of the above-mentioned control charts for autocorrelated processes are significantly affected by the sample size used in a Phase I study to estimate the control chart parameters. In this study, we investigate the effect of Phase I sample size on the run length performance of these four charts for monitoring the changes in the mean of an autocorrelated process, namely an AR(1) process. A discussion of the practical implications of the results and suggestions on the sample size requirements for effective process monitoring are provided.     

CUSUM control schemes for monitoring the covariance matrix of multivariate time series

Modified cumulative sum (CUSUM) control charts and CUSUM schemes for residuals are suggested to detect changes in the covariance matrix of multivariate time series. Several properties of these schemes are derived when the in-control process is a stationary Gaussian process. A Monte Carlo study reveals that the proposed approaches show similar or even better performance than the schemes based on the multivariate exponentially weighted moving average (MEWMA) recursion. We illustrate how the control procedures can be applied to monitor the covariance structure of developed stock market indices.     

Robustness to non-normality and autocorrelation of individuals control charts

This paper studies the effects of non-normality and autocorrelation on the performances of various individuals control charts for monitoring the process mean and/or variance. The traditional Shewhart X chart and moving range (MR) chart are investigated as well as several types of exponentially weighted moving average (EWMA) charts and combinations of control charts involving these EWMA charts. It is shown that the combination of the X and MR charts will not detect small and moderate parameter shifts as fast as combinations involving the EWMA charts, and that the performana of the X and MR charts is very sensitive to the normality assumption. It is also shown that certain combinations of EWMA charts can be designed to be robust to non-normality and very effective at detecting small and moderate shifts in the process mean and/or variance. Although autocorrelation can have a significant effect on the in-control performances of these combinations of EWMA charts, their relative out-of-control performances under independence are generally maintained for low to moderate levels of autocorrelation.