Distribution‐free Phase II CUSUM Control Chart for Joint Monitoring of Location and Scale

Chowdhury, Shovan; Mukherjee, Amitava; Chakraborti, S.

doi:10.1002/qre.1677

Cited by 79 publications

(61 citation statements)

References 39 publications

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“…When k 1 = 0,

S_{h}^{(1)}

is a frequency vector with its j th element denoting the observed count of A 1 ( h ) = j at time h and

S_{h}^{(2)}

equals ( hg 1,1 , hg 1,2 , …, hg 1, p )′ where hg 1, j is the expected count of A 1 ( h ) = j at time h . Thus, y h becomes the conventional Pearson's χ 2 statistic,

true {true\sum}_{j = 1}^{p} \frac{{((), S_{h, j}^{(1)} - S_{h, j}^{(2)})}^{2}}{S_{h, j}^{((), 2)}}

, which approximately follows

χ_{p - 1}^{2}

as h is large . In addition, we realize that y h is equal to Q h − k 1 as proved in .…”

Section: The Rank‐based Multivariate Cusum Procedures (Original Approach)mentioning

confidence: 60%

“…The following two null hypotheses involved in the multivariate process control are then considered:

H_{0}^{(1)} : μ_{1} = μ_{2} = \dots = μ_{p}, H_{0}^{(2)} : true {true\sum}_{j = 1}^{p} μ_{j} = 0,

where μ 1 , μ 2 , …, μ p are the means of the in‐control process variables. Based on the theorems, the null hypothesis,

H_{0}^{(1)} : μ_{1} = μ_{2} = \dots = μ_{p}

, for detecting the mean shift is equivalent to

H_{0}^{((), 1) *} : The probability distribution of A_{1} ((), h) is ({}, g_{1, j}, j = 1, 2, \dots, p) .

…”

Section: The Rank‐based Multivariate Cusum Procedures (Original Approach)mentioning

confidence: 99%

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A New Rank‐based Multivariate CUSUM Approach for Monitoring the Process Mean

Hwang

2015

Quality & Reliability Eng

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In many cases, data do not follow a specific probability distribution in practice. As a result, a variety of distribution-free control charts have been developed to monitor changes in the processes. An existing rank-based multivariate cumulative sum (CUSUM) procedure based on the antirank vector does not quickly detect the large shift levels of the process mean. In this paper, we explore and develop an improved version of the existing rank-based multivariate CUSUM procedure in order to overcome the difficulty. The numerical experiments show that the proposed approach dramatically outperforms the existing rank-based multivariate CUSUM procedure in terms of the out-of-control average run length. In addition, the proposed approach particularly resolves the critical problem of the original approach, which occurs in the simultaneous shifts whose components are all the same but not 0. We believe that the proposed approach can be utilized for monitoring real data.

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“…When k 1 = 0,

S_{h}^{(1)}

is a frequency vector with its j th element denoting the observed count of A 1 ( h ) = j at time h and

S_{h}^{(2)}

equals ( hg 1,1 , hg 1,2 , …, hg 1, p )′ where hg 1, j is the expected count of A 1 ( h ) = j at time h . Thus, y h becomes the conventional Pearson's χ 2 statistic,

true {true\sum}_{j = 1}^{p} \frac{{((), S_{h, j}^{(1)} - S_{h, j}^{(2)})}^{2}}{S_{h, j}^{((), 2)}}

, which approximately follows

χ_{p - 1}^{2}

as h is large . In addition, we realize that y h is equal to Q h − k 1 as proved in .…”

Section: The Rank‐based Multivariate Cusum Procedures (Original Approach)mentioning

confidence: 60%

“…The following two null hypotheses involved in the multivariate process control are then considered:

H_{0}^{(1)} : μ_{1} = μ_{2} = \dots = μ_{p}, H_{0}^{(2)} : true {true\sum}_{j = 1}^{p} μ_{j} = 0,

where μ 1 , μ 2 , …, μ p are the means of the in‐control process variables. Based on the theorems, the null hypothesis,

H_{0}^{(1)} : μ_{1} = μ_{2} = \dots = μ_{p}

, for detecting the mean shift is equivalent to

H_{0}^{((), 1) *} : The probability distribution of A_{1} ((), h) is ({}, g_{1, j}, j = 1, 2, \dots, p) .

…”

Section: The Rank‐based Multivariate Cusum Procedures (Original Approach)mentioning

confidence: 99%

A New Rank‐based Multivariate CUSUM Approach for Monitoring the Process Mean

Hwang

2015

Quality & Reliability Eng

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“…Moreover, Graham, Chakraborti, and Mukherjee (2014) outlined that 'nonparametric charts are often more robust and efficient under some heavy-tailed symmetric and skewed distributions'. Interested readers may see, Chakraborti and Graham (2007); Chakraborti, Human, and Graham (2011); Graham, Mukherjee, and Chakraborti (2012); Mukherjee, Graham, and Chakraborti (2013); Balakrishnan, Paroissin, and Turlot (2015) and Chowdhury, Mukherjee, and Chakraborti (2015) for various details on the nonparametric control charts. For further readings, we suggest the book by Qiu (2014).…”

Section: Please Scroll Down For Articlementioning

confidence: 98%

Comparisons of Shewhart-type rank based control charts for monitoring location parameters of univariate processes

Mukherjee

Sen

2015

International Journal of Production Research

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The nonparametric (distribution-free) control charts are robust alternatives to the conventional parametric control charts when the form of underlying process distribution is unknown or complicated. In this paper, we consider two new nonparametric control charts based on the Hogg-Fisher-Randle (HFR) statistic and the Savage rank statistic. These are popular statistics for testing location shifts, especially in right-skewed densities. Nevertheless, the control charts based on these statistics are not studied in quality control literature. In the current context, we study phase-II Shewhart-type charts based on the HFR and Savage statistics. We compare these charts with the Wilcoxon rank-sum chart in terms of false alarm rate, out-of-control average run-length and other run length properties. Implementation procedures and some illustrations of these charts are also provided. Numerical results based on Monte Carlo analysis show that the new charts are superior to the Wilcoxon rank-sum chart for a class of non-normal distributions in detecting location shift. New charts also provide better control over false alarm when reference sample size is small.

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“…Normality is a typical assumption needed for parametric charts, while non-parametric charts are free from any such constraints. Reference can be made for further check out to Chakraborti et al [1,2], Chowdhury et al [3], and Mukherjee and Sen [4] in the literature. Moreover, a traditional approach used in SPC is to monitor each parameter separately; however, simultaneous monitoring of more than one parameter is also becoming popular in industry.…”

Section: Introductionmentioning

confidence: 99%