X¯ Chart with Estimated Parameters: The Conditional <i>ARL</i> Distribution and New Insights

Jardim, Felipe Schoemer; Chakraborti, S.; Epprecht, Eugenio K.

doi:10.1111/poms.12985

Cited by 22 publications

(17 citation statements)

References 32 publications

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“…This choice is justified by the recommendation of Mahmoud et al of using either S p , S p / c 4 , or c 4 S p (where c 4 has argument m ( n − 1)+1) instead of the estimator

\overset{S}{true} / c_{4}

(where c 4 has argument n ) that is the usual practice. We stick thus to S p for simplicity, as did recent authors on this topic of the effect of parameter estimation on control charts performance (eg, Castagliola et al, Epprecht et al, Faraz et al, Jardim et al, Loureiro et al, and Guo and Wang). The estimator S p is given by the following:

S_{p} = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} S_{i}^{2}} = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} ((), \frac{1}{n - 1} \sum_{j = 1}^{n} {(X_{italicij} - {\overset{X}{true}}_{i})}^{2})} .

…”

Section: Background and Underlying Assumptionsmentioning

confidence: 99%

“…Jardim et al derived the c.d.f. of conditional ARL 0 ( CARL 0 ); in this case:

F_{{italicCARL}_{0}} ((), t) = P ((), {CARL}_{0} < t) = \int_{- \infty}^{\infty} ([], F_{χ_{m (n - 1)}^{2}} ((), \frac{m ((), n - 1)}{L^{2}} F_{χ_{1, [\frac{d^{2}}{m}]}^{2}}^{- 1} ((), 1 - \frac{1}{t})) f_{D} ((), d)),

where

χ_{1, ([], \frac{d^{2}}{m})}^{2}

denotes a noncentral chi‐square variable with 1 degree of freedom and noncentrality parameter d 2 / m and the generic notation

F_{U}^{- 1} (() \cdot)

represents the inverse of the c.d.f.…”

Section: Background and Underlying Assumptionsmentioning

confidence: 99%

“…Focusing on different control charts, several authors have recently used the condition in Equation to determine, for given values of n , the minimum number of phase I samples ( m *) that satisfies that condition. Given that the values of m * found were often unpractically large, authors such as Albers and Kallenberg, 24 Gandy and Kvaløy, Faraz et al, Goedhart et al, Guo and Wang, and Jardim et al, proposed an adjustment to the control limit factor ( L *) that ensures the satisfaction of condition in Equation with a specified number (smaller than m *) of phase 1 samples. As a consequence of the use of the adjusted limit, there will be a deterioration of the OOC performance of the control chart.…”

Section: Background and Underlying Assumptionsmentioning

confidence: 99%

“…Considering the

\overset{X}{true}

chart, Saleh et al have recommended the numbers of phase I samples that limit the standard deviation of the conditional ARL 0 to a specified value; at the same time, Epprecht et al recommended the numbers of phase I samples that limit to a specified value the probability that the conditional false‐alarm rate of an S chart exceeded some maximum tolerated value specified by its user. Later, a considerable number of papers appeared that analyze the effect of parameter estimation on the conditional IC performance of several types of control charts (eg, Saleh et al, Faraz et al, Goedhart et al, Jardim et al, Loureiro et al, and Guo and Wang). A common finding of those works is that the number of samples required is orders of magnitude larger than the classic recommendation of 20 to 30 samples (see, for example, Montogomery), and often unpractical.…”

Section: Introductionmentioning

confidence: 99%

“…As a response to this problem, some authors proposed adjustments to the control limits of the charts that would enable achieving acceptable IC performance with practicable numbers of phase I samples. See, for example, Gandy and Kvaløy, Faraz et al, Saleh et al, Goedhart et al, Jardim et al, and Guo and Wang . An overview and comparison of different proposals of adjustments to the limits of the

\overset{X}{true}

chart is given in Jardim et al…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Design of an X̅ control chart with guaranteed joint in‐control and out‐of‐control performances when the mean and variance are estimated

Mosquera

Aparisi

Epprecht

2019

Quality & Reliability Eng

Self Cite

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During phase I analysis, the mean and variability of the process are estimated, in order to obtain the control limits for the chart. The effect of the estimation errors in the performance of the chart (phase II) has been widely studied in the literature. However, the papers have been mainly orientated to the in‐control performance of the chart. In this paper, the in‐control and out‐of‐control performances of the trueX¯ control chart are considered at the same time. Therefore, the number of samples to be taken in the phase I analysis is calculated, taking into account the possible deterioration of both metrics. In addition, an adjustment in the control limit is proposed, to reduce this number of samples.

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“…This choice is justified by the recommendation of Mahmoud et al of using either S p , S p / c 4 , or c 4 S p (where c 4 has argument m ( n − 1)+1) instead of the estimator

\overset{S}{true} / c_{4}

S_{p} = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} S_{i}^{2}} = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} ((), \frac{1}{n - 1} \sum_{j = 1}^{n} {(X_{italicij} - {\overset{X}{true}}_{i})}^{2})} .

…”

Section: Background and Underlying Assumptionsmentioning

confidence: 99%

“…Jardim et al derived the c.d.f. of conditional ARL 0 ( CARL 0 ); in this case:

F_{{italicCARL}_{0}} ((), t) = P ((), {CARL}_{0} < t) = \int_{- \infty}^{\infty} ([], F_{χ_{m (n - 1)}^{2}} ((), \frac{m ((), n - 1)}{L^{2}} F_{χ_{1, [\frac{d^{2}}{m}]}^{2}}^{- 1} ((), 1 - \frac{1}{t})) f_{D} ((), d)),

where

χ_{1, ([], \frac{d^{2}}{m})}^{2}

denotes a noncentral chi‐square variable with 1 degree of freedom and noncentrality parameter d 2 / m and the generic notation

F_{U}^{- 1} (() \cdot)

represents the inverse of the c.d.f.…”

Section: Background and Underlying Assumptionsmentioning

confidence: 99%

Section: Background and Underlying Assumptionsmentioning

confidence: 99%

“…Considering the

\overset{X}{true}

Section: Introductionmentioning

confidence: 99%

\overset{X}{true}

chart is given in Jardim et al…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Design of an X̅ control chart with guaranteed joint in‐control and out‐of‐control performances when the mean and variance are estimated

Mosquera

Aparisi

Epprecht

2019

Quality & Reliability Eng

Self Cite

View full text Add to dashboard Cite

show abstract

Finite Horizon Process Monitoring

Chakraborti

Celano

Castagliola

et al. 2021

Wiley StatsRef: Statistics Reference Online

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In some industrial environments flexibility is a key for the manufacturer, that is, the ability to switch frequently from the production of one product code to another and produce batches of different items during consecutive short production time intervals, called finite production horizons; such a setting is called a finite horizon process (FHP). The frequent process changeovers may require new setup activities and machine reconfigurations every time before the beginning of a new production run. An erroneous setup or a process failure can negatively impact the quality of the manufactured products. In order to perform statistical quality control for these types of processes, practitioners have to deal with the challenging issues of designing, scheduling, and performing only a limited number of inspections. This is typically not the case in traditional quality control and monitoring applications. In this article, we highlight various aspects of statistical process monitoring (SPM) implemented in an FHP setting that need to be considered by quality practitioners. We describe available statistical process monitoring tools, such as the control charts, properly designed to monitor quality characteristics in the FHP setting. Finally, we discuss some of the still open challenges in the area of FHP monitoring.

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Unconditional performance of the X¯ chart: Comparison among five standard deviation estimators

Sobue

Jardim

Camargo

et al. 2020

Quality & Reliability Eng

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To design the control chart, the in‐control process mean and standard deviation must be estimated from historical samples, negatively affecting the chart's performance. The grand mean of a sample is the well‐established estimator for the process mean. However, regarding the standard deviation, the chart's user has at least five different estimators available in the literature to choose from. In this paper, using intensive simulations, we study and compare the performance of the chart (under normality assumption and three‐sigma limits) among the most five used standard deviation estimators. The unconditional in‐control run length (RL0) is the most commonly used performance measure of a control chart, and many authors base their comparisons on the expectation of the RL0 or on the mean square error of the estimators. In contrast, we based our comparison on the proportions of the RL0 concentrated at some intervals that are usually considered undesired by the practitioner due to the high incidence of false alarms during the process monitoring (e.g., RL0 between 1 and 200), occurring earlier than expected when compared with known parameter situations. From our results, all the studied standard deviation estimators generate similar performances. However, even with this similarity, we showed that one of the most recommended standard deviation estimators in the literature in control charts with estimated parameters may be the most inappropriate choice based on the undesired RL0 proportions.

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X¯ Chart with Estimated Parameters: The Conditional ARL Distribution and New Insights

Cited by 22 publications

References 32 publications

Design of an X̅ control chart with guaranteed joint in‐control and out‐of‐control performances when the mean and variance are estimated

Design of an X̅ control chart with guaranteed joint in‐control and out‐of‐control performances when the mean and variance are estimated

Finite Horizon Process Monitoring

Unconditional performance of the X¯ chart: Comparison among five standard deviation estimators

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