The Run Length Distribution of the CUSUM with Estimated Parameters

Journal of Quality Technology

Volumn 36, Issue 1, 2004, Pages 95-108

  Jones, L Allison ^a,d   Champ, Charles W ^b,d   Rigdon, Steven E ^c,d  

^a AUBURN UNIVERSITY   (United States)

^b GEORGIA SOUTHERN UNIVERSITY   (United States)

^c SOUTHERN ILLINOIS UNIVERSITY EDWARDSVILLE   (United States)

^d Department of Management ^*  (United States)

Author keywords

Average Run Length; Control Charts; Cumulative Sum

Indexed keywords

APPROXIMATION THEORY; COMPUTATIONAL METHODS; EVALUATION; MONITORING; PARAMETER ESTIMATION; PERFORMANCE; QUALITY CONTROL; STATISTICAL METHODS;

AVERAGE RUN LENGTHS; CONTROL CHARTS; CUMULATIVE SUM;

PRODUCTION ENGINEERING;

EID: 0442295421     PISSN: 00224065     EISSN: None     Source Type: Journal    
DOI: 10.1080/00224065.2004.11980254     Document Type: Article

References (31)

1. BAGSHAW, M. and JOHNSON, R. A. (1975). "The Influence of Reference Values and Estimated Variance on the ARL of CUSUM Tests". Journal of the Royal Statistical Society B 37, pp. 413-420.
2. BROOK, D. and EVANS, D. A. (1972). "An Approach to the Probability Distribution of CUSUM Run Length". Biometrika 59. pp. 539-549.
3. BURROUGHS, T. E.; RIGDON, S. E.; and CHAMP, C. W. (1993). "An Analysis of Shewhart Charts with Runs Rules When No Standards Are Given". Proceedings of the Quality and Productivity Section of the American Statistical Association August 8-12, San Francisco, CA, pp. 16-19.
4. BURROUGHS, T. E.; RIGDON, S. E.; and CHAMP, C. W. (1995). "Analysis of the Shewhart S-Chart with Runs Rules When No Standards Are Given". Proceedings of the Twenty-Sixth Annual Meeting of the Midwest Decision Sciences Institute May 4-6, St. Louis, MO, pp. 268-270.
5. CHAMP, C. W.; RIGDON, S. E.; and SCHARNAGL, K. A. (2001). "A Method for Deriving Integral Equations Useful in Control Chart Performance Analysis". Nonlinear Analysis 47, pp. 2089-2101.
6. CHAKRABORTI, S. (2000). "Run Length, Average Run Length, and False Alarm Rate of Shewhart X-bar Chart: Exact Derivations by Conditioning". Communications in Statistics: Simulation and Computation 29, pp. 61-81.
7. CHEN, G. (1997). "The Mean and Standard Deviation of the Run Length Distribution of X-bar Charts when Control Limits are Estimated". Statistica Sinica 7, pp. 789-798.
8. DEL CASTILLO, E. (1996a). "Run Length Distributions and Economic Design of Charts with Unknown Process Variance". Metrika 43, pp. 189-201.
9. DEL CASTILLO, E. (1996b). "Evaluation of Run Length Distribution for Charts with Unknown Variance". Journal of Quality Technology 28, pp. 116-122.
10. DERMAN, C. and Ross, S. (1995). "An Improved Estimator of σ in Quality Control". Probability in the Engineering and Informational Sciences 9, pp. 411-415.
11. FELLNER, W. H. (1990). "Algorithm AS 258: Average Run Lengths for Cumulative Sum Schemes". Applied Statistics 39, pp. 402-412.
12. GAN, F. F. (1991). "An Optimal Design of CUSUM Quality Control Charts". Journal of Quality Technology 23, pp. 279-286.
13. GAN, F. F. (1993). "The Run Length Distribution of a Cumulative Sum Control Chart". Journal of Quality Technology 25, pp. 205-215.
14. GHOSH, B. K.; REYNOLDS, M. R., JR.; and HUI, Y. V. (1981). "Shewhart -Charts with Estimated Process Variance". Communications in Statistics - Theory and Methods 18, pp. 1797-1822.
15. GOEL, A. L. and WU, S. M. (1971). "Determination of ARL and a Contour Nomogram for CUSUM Charts to Control Normal Mean". Technometrics 13, pp. 221-230.
16. GOLD, M. S. (1989). "The Geometric Approximation to the CUSUM Run Length Distribution". Biometrika 76, pp. 725-733.
17. HAWKINS, D. M. and OLWELL, D. H. (1998). Cumulative Sum Charts and Charting for Quality Improvement. Springer. New York, NY.
18. JONES, L. A. (2002). "The Statistical Design of EWMA Control Charts with Estimated Parameters". Journal of Quality Technology 34, pp. 277-288.
19. JONES, L. A.; CHAMP, C. W.; and RIGDON, S. E. (2001). "The Performance of Exponentially Weighted Moving Average Charts with Estimated Parameters". Technometrics 43, pp. 156-167.
20. KEMP, K. W. (1961). "The Average Run Length of the Cumulative Sum Chart when a V-Mask is Used". Journal of the Royal Statistical Society B 23, pp. 149-153.
21. LU, C. W. and REYNOLDS, M. R., JR. (1999). "EWMA Control Charts for Monitoring the Mean of Autocorrelated Processes". Journal of Quality Technology 31, pp. 166-188.
22. LUCẼNO, A. and PUIG-PEY, J. (2000). "Evaluation of the Run-Length Probability Distribution for CUSUM Charts: Assessing Chart Performance". Technometrics 42. pp. 411-416.
23. LUCẼNO, A. and PUIG-PEY, J. (2002). "Computing the Run Length Probability Distribution for CUSUM Charts". Journal of Quality Technology 34, pp. 209-215.
24. NG, C. H. and CASE, K. E. (1992). "Control Limits and the ARL: Some Surprises". First Industrial Engineering Research Conference Proceedings, pp. 127-129.
25. PAGE, E. S. (1954). "Continuous Inspection Schemes". Biometrika 41, pp. 100-114.
26. QUESENBERRY, C. P. (1993). "The Effect of Sample Size on Estimated Limits for and Control Charts". Journal of Quality Technology 25, pp. 237-247.
27. VAN DOBBEN DE BRUYN, C. S. (1968). Cumulative Sum Tests: Theory and Practice, Griffin, London, UK.
28. WOODALL, W. H. (1983). "The Distribution of the Run Length of One-Sided CUSUM Procedures for Continuous Random Variables". Technometrics 25, pp. 295-301.
29. WOODALL, W. H. (1986). "The Design of CUSUM Quality Control Charts". Journal of Quality Technology 18, pp. 99-102.
30. YANG, M. C. K. (1990). "Properties of the CUSUM Chart When There is No Standard Value (in Chinese)". Journal of Chinese Statistical Association 28, pp. 57-77.
31. YANG, Z.; XIE, M.; KURALMANI, V.; and TSUI, K. (2002). "On the Performance of Geometric Charts with Estimated Control Limits". Journal of Quality Technology 34, pp. 448-458.