A Comparison of the Performance of Phase II Simple Linear Profile Control Charts when Parameters are Estimated

Because of the characteristics of a system or process, several prespecified changes may happen in some statistical process control applications. Thus, one possible and challenging problem in profile monitoring is detecting changes away from the ‘normal’ profile toward one of several prespecified ‘bad’ profiles. In this article, to monitor the prespecified changes in linear profiles, two two‐sided cumulative sum (CUSUM) schemes are proposed based on Student's t‐statistic, which use two separate statistics and a single statistic, respectively. Simulation results show that the CUSUM scheme with a single statistic uniformly outperforms that with two separate statistics. Besides, both CUSUM schemes perform better than alternative methods in detecting small shifts in prespecified changes, and become comparable on detecting moderate or large shifts when the number of observations in each profile is large. To overcome the weakness in the proposed CUSUM methods, two modified CUSUM schemes are developed using z‐statistic and studied when the in‐control parameters are estimated. Simulation results indicate that the modified CUSUM chart with a single charting statistic slightly outperforms that with two separate statistics in terms of the average run length and its standard deviation. Finally, illustrative examples indicate that the CUSUM schemes are effective. Copyright © 2016 John Wiley & Sons, Ltd.

Section: Modified Cusum Schemes With Estimated Parametersmentioning

confidence: 99%

{\overset{true¯}{\overset{β}{true}}}_{0} = \frac{\sum_{j = 1}^{m} {\overset{β}{true}}_{0 j}}{m}, {\overset{true¯}{\overset{β}{true}}}_{1} = \frac{\sum_{j = 1}^{m} {\overset{β}{true}}_{1 j}}{m} and M S E = \frac{\sum_{j = 1}^{m} M S E_{j}}{m},

where

{\overset{true^}{β}}_{0 j}

and

{\overset{true^}{β}}_{1 j}

are the least squares estimators of the linear profile parameters β 0 j and β 1 j , and MSE j is an estimator of the variance

σ_{ε 0 j}^{2}

. These estimators are defined as follows:

\begin{matrix} left \\ {\overset{β}{true}}_{0 j} = {\overset{Y}{true}}_{j} - {\overset{β}{true}}_{1 j} \overset{true¯}{x}, {\overset{β}{true}}_{1 j} = S_{x y (j)} / S_{x x}, {\overset{Y}{true}}_{j} = n^{- 1} \sum_{i = 1}^{n} y_{i j}, \overset{true¯}{x} = n^{- 1} \sum_{i = 1}^{n} x_{i}, \end{matrix}

…”

Section: Modified Cusum Schemes With Estimated Parametersmentioning

confidence: 99%

CUSUM Schemes for Monitoring Prespecified Changes in Linear Profiles

Zhang

Shang

et al. 2016

“…In addition, the EWMA control chart with estimated parameters was studied by Saleh et al . Moreover, Aly et al . compared the performance of different simple linear profile monitoring approaches using the SDARL metric.…”

Section: Introductionmentioning

confidence: 83%

A Reevaluation of the Adaptive Exponentially Weighted Moving Average Control Chart When Parameters are Estimated

Aly

Saleh

Mahmoud

et al. 2014

Self Cite

The performance of control charts can be adversely affected when based on parameter estimates instead of known in‐control parameters. Several studies have shown that a large number of phase I observations may be needed to achieve the desired in‐control statistical performance. However, practitioners use different phase I samples and thus different parameter estimates to construct their control limits. As a consequence, there would be in‐control average run length (ARL) variation between different practitioners. This kind of variation is important to consider when studying the performance of control charts with estimated parameters. Most of the previous literature has relied primarily on the expected value of the ARL (AARL) metric in studying the performance of control charts with estimated parameters. Some recent studies, however, considered the standard deviation of the ARL metric to study the performance of control charts. In this paper, the standard deviation of the ARL metric is used to study the in‐control and out‐of‐control performance of the adaptive exponentially weighted moving average (AEWMA) control chart. The performance of the AEWMA chart is then compared with that of the Shewhart trueX¯ and EWMA control charts. The simulation results show that the AEWMA chart might represent a good solution for practitioners to achieve a reasonable amount of ARL variation from the desired in‐control ARL performance. In addition, we apply a bootstrap‐based design approach that provides protection against frequent false alarms without deteriorating too much the out‐of‐control performance. Copyright © 2014 John Wiley & Sons, Ltd.

“…compared the performance of the Shewhart

\overset{X}{true} prefix-

, EWMA and the univariate AEWMA charts using this metric. It was also used by Aly et al . to compare the performance of different simple linear profile monitoring approaches.…”

Section: Introductionmentioning

confidence: 99%

The Performance of the Multivariate Adaptive Exponentially Weighted Moving Average Control Chart with Estimated Parameters

Aly

Mahmoud

Hamed

2015

Self Cite

The multivariate adaptive exponentially weighted moving average control chart (MAEWMA) can detect shifts of different sizes while diminishing the inertia problem to a large extent. Although it has several advantages compared to various multivariate charts, previous literature has not considered its performance when the parameters are estimated. In this study, the performance of the MAEWMA chart with estimated parameters is studied while considering the practitioner-topractitioner variation. This kind of variation occurs due to using different Phase I samples by different practitioners in estimating the unknown parameters. The simulation results in this paper show that estimating the parameters results in extensively excessive false alarms and as a result a large number of Phase I samples is needed to achieve the desired incontrol performance. Using small number of Phase I samples in estimating the parameters may result in an in-control ARL distribution that almost completely lies below the desired value. To handle this problem, we strongly recommend the use of a bootstrap-based algorithm to adjust the control limit of the MAEWMA chart. This algorithm enables practitioners to achieve, with a certain probability, an in-control ARL that is greater than or equal to the desired value while using the available number of Phase I samples.