A HEWMA-CUSUM control chart for the Weibull distribution

The Weibull distribution is one of the most popular distributions for lifetime modeling. However, there has not been much research on control charts for a Weibull distribution. Shewhart control is known to be inefficient to detect a small shift in the process, while exponentially weighted moving average (EWMA) and cumulative sum control chart (CUSUM) charts have the ability to detect small changes in the process. To enhance the performance of a control chart for a Weibull distribution, we introduce a new control chart based on hybrid EWMA and CUSUM statistic, called the HEWMA-CUSUM chart. The performance of the proposed chart is compared with the existing chart in terms of the average run length (ARL). The proposed chart is found to be more sensitive than the existing chart in ARL. A simulation study is provided for illustration purposes. A real data is also applied to the proposed chart for practical use.     

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 random intrinsic fast initial response of one-sided CUSUM charts

This article analyses the performance of a one-sided cumulative sum (CUSUM) chart that is initialized using a random starting point following the natural or intrinsic probability distribution of the CUSUM statistic. By definition, this probability distribution remains stable as the chart is used. The probability that the chart starts at zero according to this intrinsic distribution is always smaller than one, which confers on the chart a fast initial response feature. The article provides a fast and accurate algorithm to compute the in-control and out-of-control average run lengths and run-length probability distributions for one-sided CUSUM charts initialized using this random intrinsic fast initial response (RIFIR) scheme. The algorithm also computes the intrinsic distribution of the CUSUM statistic and random samples extracted from this distribution. Most importantly, no matter how the chart was initialized, if no level shifts and no alarms have occurred before time τ?>?0, the distribution of the run length remaining after τ is provided by this algorithm very accurately, provided that τ is not too small.     

A mixed GWMA–CUSUM control chart for monitoring the process mean

The memory-type control charts are widely used in the process and service industries for monitoring the production processes. The reason is their sensitivity to quickly react against the small process disturbances. Recently, a new cumulative sum (CUSUM) chart has been proposed that uses the exponentially weighted moving average (EWMA) statistic, called the EWMA–CUSUM chart. Similarly, in order to further enhance the sensitivity of the EWMA–CUSUM chart, we propose a new CUSUM chart using the generally weighted moving average (GWMA) statistic, called the GWMA–CUSUM chart, for efficiently monitoring the process mean. The GWMA–CUSUM chart encompasses the existing CUSUM and EWMA–CUSUM charts. Extensive Monte Carlo simulations are used to explore the run length profiles of the GWMA–CUSUM chart. Based on comprehensive run length comparisons, it turns out that the GWMA–CUSUM chart performs substantially better than the CUSUM, EWMA, GWMA, and EWMA–CUSUM charts when detecting small shifts in the process mean. An illustrative example is also presented to explain the implementation and working of the EWMA–CUSUM and GWMA–CUSUM charts.     

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.     

A non parametric CUSUM control chart based on the Mann–Whitney statistic

We consider a novel univariate non parametric cumulative sum (CUSUM) control chart for detecting the small shifts in the mean of a process, where the nominal value of the mean is unknown but some historical data are available. This chart is established based on the Mann–Whitney statistic as well as the change-point model, where any assumption for the underlying distribution of the process is not required. The performance comparisons based on simulations show that the proposed control chart is slightly more effective than some other related non parametric control charts.     

Computation of the run-length percentiles of CUSUM control charts under changes in variances

In this paper, the run-length distributions of cumulative sum (CUSUM) charts for monitoring mean changes under normal distributions have been investigated thoroughly. However, there are few studies devoted to the analysis of the run-length distributions of CUSUM charts under changes in process variances. Motivated by this, this paper develops a fast and accurate algorithm based on piecewise collocation method for computing the run-length distributions of CUSUM scale charts. It is shown that the proposed method can provide more accurate approximation to the run-length distribution than the conventional Gauss-type quadrature-based methods applied to the CUSUM location charts. Some computational aspects for facilitating computation load are discussed, including the alternative formulation based on matrix decomposition and the geometric approximation to the distribution of large run lengths.     

A control chart for variance based on squared ranks

A nonparametric control chart for variance is proposed. The chart is constructed following the change-point approach through the recursive use of the squared ranks test for variance. It is capable of detecting changes in the behaviour of individual observations with performance similar to a self-starting CUSUM chart for scale when normality is assumed, and a relatively better power when assessing nonnormal observations. A comparison is also made with two equivalent nonparametric charts based on Mood and Ansari-Bradley statistics. When dealing with symmetrical distributions, the proposed chart shows smaller (better) out-of-control average run length (ARL), and a competing performance otherwise. In addition, sensitivity to changes in mean and variance at the same time was tested. Extensive Monte Carlo simulation was used to measure performance, and a practical example is provided to illustrate how the proposed control chart can be implemented in practice.     

On increasing the sensitivity of mixed EWMA–CUSUM control charts for location parameter

Control chart is an important statistical technique that is used to monitor the quality of a process. Shewhart control charts are used to detect larger disturbances in the process parameters, whereas cumulative sum (CUSUM) and exponential weighted moving average (EWMA) are meant for smaller and moderate changes. In this study, we enhanced mixed EWMA–CUSUM control charts with varying fast initial response (FIR) features and also with a runs rule of two out of three successive points that fall above the upper control limit. We investigate their run-length properties. The proposed control charting schemes are compared with the existing counterparts including classical CUSUM, classical EWMA, FIR CUSUM, FIR EWMA, mixed EWMA–CUSUM, 2/3 modified EWMA, and 2/3 CUSUM control charting schemes. A case study is presented for practical considerations using a real data set.     

A Nonparametric Synthetic Control Chart Using Sign Statistic

This article presents a synthetic control chart for detection of shifts in the process median. The synthetic chart is a combination of sign chart and conforming run-length chart. The performance evaluation of the proposed chart indicates that the synthetic chart has a higher power of detecting shifts in process median than the Shewhart charts based on sign statistic as well as the classical Shewhart X-bar chart for various symmetric distributions. The improvement is significant for shifts of moderate to large shifts in the median. The robustness studies of the proposed synthetic control chart against outliers indicate that the proposed synthetic control chart is robust against contamination by outliers.     

Nonparametric quality control charts based on the sign statistic

Nonparametric control chart are presented for the problem of detecting changes in the process median (or mean), or changes in the process variability when samples are taken at regular time intervals. The proposed procedures are based on sign-test statistics computed for each sample, and are used in Shewhart and cumulative sum control charts. When the process is in control the run length distributions for the proposed nonparametric control charts do not depend on the distribution of the observations. An additional advantage of the non-parametric control charts is that the variance of the process does not need to be established in order to set up a control chart for the mean. Comparisons with the corresponding parametric control charts are presented. It is also shown that curtailed sampling plans can considerably reduce the expected number of observations used in the Shewhart control schemes based on the sign statistic.     

Using the predictive distribution to determine control limits for the Bayesian MEWMA chart

Bayesian control charts have been proposed for monitoring multivariate processes with the multivariate exponentially weighted moving average (MEWMA) statistic. It has been suggested that we use limits based on the predictive distribution of the MEWMA statistic. This analysis, however is based on the erroneous result that the average run length (ARL) is a function of the exceedance probability, that is, the probability that the first point exceeds the control limit. We show how this result can be corrected and we discuss how the Bayesian MEWMA chart with limits based on the predictive distribution compares with other multivariate control chart procedures.     

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.     

Statistical control charts for monitoring the mean of a stationary process

Recently statistical process control (SPC) methodologies have been developed to accommodate autocorrelated data. A primary method to deal with autocorrelated data is the use of residual charts. Although this methodology has the advantage that it can be applied to any autocorrelated data it needs time series modeling efforts. In addition for a X residual chart the detection capability is sometimes small compared to the X chart and EWMA chart. Zhang (1998) proposed the EWMAST chart which is constructed by charting the EWMA statistic for stationary processes to monitor the process mean. The performance of the EWMAST chart the X chart the X residual chart and other charts were compared in Zhang (1998). In this paper comparisons are made among the EWMAST chart the CUSUM residual chart and EWMA residual chart as well as the X residual chart and X chart via the average run length.     

A Markov-chain method for computing the run-length distribution of the self-starting cumulative sum scheme

We propose an analytic method for computing the run-length distribution of the cumulative sum (CUSUM) of Q statistics. The method is based on a model in which the operation of this CUSUM is embedded in a nonstationary, discrete-time Markov chain. The calculations of the method agree closely with those of Monte Carlo simulation, supporting the method's accuracy. Our results facilitate understanding the effectiveness of the CUSUM of Q statistics in detecting process mean shifts.     

Comparison of Confidence Intervals on Between Group Variance in Unbalanced Heteroscedastic One-Way Random Models

A computer algorithm for computing the alternative distributions of the Wilcoxon signed rank statistic under shift alternatives is discussed. An explicit error bound is derived for the numeric integration approximation to these distributions.

A nonparametric process control procedure in which the standard CUSUM procedure is applied to the Wilcoxon signed rank statistic is discussed. In order to implement this procedure, the distribution of the Wilcoxon statistic under shift of the underlying distribution from its point of symmetry needs to be computed. The average run length of the nonparametric and parametric CUSUM are compared.     

Nonparametric control chart based on change-point model

A change-point control chart for detecting shifts in the mean of a 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 a nonparametric chart based on the Mann–Whitney statistic for a change in mean and adapted for repeated sequential use. We do not require any knowledge of the underlying distribution such as the normal assumption. Particularly, this distribution robustness could be a significant advantage in start-up or short-run situations where we usually do not have knowledge of the underlying distribution. The simulated results show that our approach has a good performance across the range of possible shifts and it can be used during the start-up stages of the process.      

Conditional design of the CUSUM median chart for the process position when process parameters are unknown

In practice, different practitioners will use different Phase I samples to estimate the process parameters, which will lead to different Phase II control chart's performance. Researches refer to this variability as between-practitioners-variability of control charts. Since between-practitioners-variability is important in the design of the CUSUM median chart with estimated process parameters, the standard deviation of average run length (SDARL) will be used to study its properties. It is shown that the CUSUM median chart requires a larger amount of Phase I samples to sufficiently reduce the variation in the in-control ARL of the CUSUM median chart. Considering the limitation of the amount of the Phase I samples, a bootstrap approach is also used here to adjust the control limits of the CUSUM median chart. Comparisons are made for the CUSUM and Shewhart median charts with estimated parameters when using the adjusted- and unadjusted control limits and some conclusions are made.