Determination of A.R.L. and a Contour Nomogram for Cusum Charts to Control Normal Mean

Goel, Amrit L.; Wu, S. M.

doi:10.1080/00401706.1971.10488778

Cited by 54 publications

(29 citation statements)

References 10 publications

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“…The numerical resolution of Fredholm equations is a very large topic in applied mathematics and is beyond the scope of this paper. For the ARL evaluation of a detection scheme, the paper of Goel and Wu [11] can be mentioned.…”

Section: Analytical Approximationmentioning

confidence: 99%

Change detection for uncertain autoregressive dynamic models through nonparametric estimation

Hilgert

Verdier

Vila

2016

Statistical Methodology

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A new statistical approach for on-line change detection in uncertain dynamic system is proposed. In change detection problem, the distribution of a sequence of observations can change at some unknown instant. The goal is to detect this change, for example a parameter change, as quickly as possible with a minimal risk of false detection. In this paper, the observations come from an uncertain system modeled by an autoregressive model containing an unknown functional component. The popular Page's CUSUM rule is not applicable anymore since it requires the full knowledge of the model. A new detection CUSUM-like scheme is proposed, which is based on the nonparametric estimation of the unknown component from a learning sample. Moreover, the estimation procedure can be updated on line which ensures a better detection, especially at the beginning of the monitoring procedure. Simulation trials were performed on a model describing a water treatment process and show the interest of this new procedure with respect to the classic CUSUM rule.

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Section: Analytical Approximationmentioning

confidence: 99%

Change detection for uncertain autoregressive dynamic models through nonparametric estimation

Hilgert

Verdier

Vila

2016

Statistical Methodology

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“…Goel and Wu (1971) solved the characteristic equation for CUSUM chart for controling process mean, numerically by using Gaussian Hermite quadrature. In this section, we show the algorithms to solve the characteristic integral Equations (2.1) and (2.2) for variance CUSUM.…”

Section: Solution and Numerical Algorithmmentioning

confidence: 99%

Design of Variance CUSUM

Lee¹,

Hong²,

Lee³

2009

Korean Journal of Applied Statistics

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We suggest a fast and accurate algorithm to compute ARLs of CUSUM chart for controling process variance.The algorithm solves the characteristic integral equations of CUSUM chart (for controling variance). The algorithm is directly applicable for the cases of odd sample sizes. When the sample size is even, by using well-known approximation algorithm combinedly with the new algorithm for neighboring odd sample sizes, we can also evaluate the ARLs of CUSUM charts efficiently and accurately. Based on the new algorithm, we consider the optimal design of upward and downward CUSUM charts for controling process variance.

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“…Page (1954) first formulated the integral equation for CUSUM ARL evaluation by first-step analysis. Later, Goel and Wu (1971) and Lucas (1976) used the same integral equation but solved it with different numerical approaches. Given the distribution of the quality observations, Lorenzen and Vance (1986) evaluated the upper-sided ARL, ARL + , and lower-sided ARL, ARL − , using the following integral equations, respectively:…”

Section: Quality Impact Induced By Control Chartsmentioning

confidence: 99%

Design of EWMA and CUSUM control charts subject to random shift sizes and quality impacts

Chen

2007

IIE Transactions

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Statistical process control charts are important tools for detecting process shifts. To ensure accurate, responsive fault detection, control chart design is critical. In the literature, control charts are typically designed by minimizing the control chart's responding time, i.e., average run length (ARL), to an anticipated shift size under a tolerable false alarm rate. However, process shifts, originating from various variation sources, often come with different sizes and result in different degrees of quality impacts. In this paper, we propose a new performance measure for EWMA and CUSUM control chart design to take into consideration the variable shift sizes and corresponding quality impacts. Unlike economic designs of control charts that suffer from a complex cost structure and intensive numerical computation, the proposed design methodology does not involve any cost estimation and the design procedure is as simple as looking up tables. Given the Gaussian random shifts and quadratic quality loss function, we show that the proposed design has a significant reduction in the quality impact as compared to conventional ARL-based designs. Guidelines and useful worksheets for practical implementation of the proposed designs are then suggested to practitioners with different knowledge levels of the process excursions.

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Determination of A.R.L. and a Contour Nomogram for Cusum Charts to Control Normal Mean

Cited by 54 publications

References 10 publications

Change detection for uncertain autoregressive dynamic models through nonparametric estimation

Change detection for uncertain autoregressive dynamic models through nonparametric estimation

Design of Variance CUSUM

Design of EWMA and CUSUM control charts subject to random shift sizes and quality impacts

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