Replacing the sequence probability ratio test with the V mask in the cumulative summative control chart

Document Type : Original Article

Authors

1 Bahonar University of Kerman

2 Associate Professor, Ferdowsi University of Mashhad, Faculty of Mathematical Sciences, Department of Statistics

Abstract
Cumulative cumulative control charts are presented in most quality control books regardless of the statistical formulas behind the V-mask. In this paper, after introducing and reviewing the cumulative sum control chart, the decision rule is examined and its relationship with sequence probability ratio tests. Many references and quality control books consider the V mask method to be equivalent to a cumulative test with inverted data. Although the two methods have many similarities, their decision-making rules are not the same. In order to further clarify, in this article, the similarities and differences between the two approaches are examined and compared.

Keywords


[1] بهبودیان، ج. )۱۳۷۱ .)آمار ریاضی. انتشارات امیرکبیر.
[2] پرچمی، ع. و ماشینچی، م. )۱۳4۱ .)کنترل کیفیت آماری. انتشارات دانشگاه شهید باهنر کرمان.
[3] مقدس، س.م. و صالحاولیاء، م. )۱۳۷۱ .)کنترل کیفیت: سیستم، سازماندهی، روشهای آماری. جهاد دانشگاهی صنعتی شریف.
[4] مونتگمری )۱۳۷۶ .)کنترل کیفیت آماری. ترجمهی رسول نورالسناء. انتشارات دانشگاه علم و صنعت.
[5] نقندریان، ک. )۱۳۸9 .)کنترل کیفیت آماری. انتشارات دانشگاه علم و صنعت
[6] Armitage, P. (1950). Sequential Analysis with More than Two Alternative Hypotheses, and its Relation to Discriminant Function Analysis. Journal of the Royal Statistical Society, 12(1), 137-144.
[7] Biswas, S. (1996). Statistics of Quality Control: Sampling Inspection and Reliability. New Age International Publishers Eastern Ltd.
[8] Goel, A. L. (1982). Cumulative Sum Control Charts. Encyclopedia of Statistical Sciences. S. Kots and N. L. Johnson, Eds, Vol. 2, John Wiley & Sons, New York, 233-241.
[9] Goel, A. L. (2011). Cumulative sum control charts. In Handbook of Methods and Applications of Statistics: Engineering, Quality Control, and Physical Sciences, N Balakrishnan (ed.). John Wiley & Sons: New York, 120-129.
[10] Graham, M. A., Chakraborti, S., Mukherjee, A. (2014). Design and implementation of CUSUM exceedance control charts for unknown location. International Journal of Production Research, 52(18), 5546-5564.
[11] Grant, E. L., Leavenworth, R. S. (1988). Statistical Quality Control. 6th Ed., McGraw. New York.
[12] Johnson, N. L. (1961). A simple theoretical approach to cumulative sum control charts. Journal of the American Statistical Association, 56, 835- 840.
[13] Johnson, N. L. , & Leone, F. C. (1964). Statistics and Exprimental Design in Engineering and the Physical Science, 1, John Wiley & Sons, New York.
[14] Jones, L. A. , Champ, C. W., Rigdon, S. E. (2004). The run length distribution of the CUSUM with estimated parameters. Journal of Quality Technology, 36(1), 95-108.
[15] Lucas, J. M. , & Crosier R. B. (1982). Fast initial response for CUSUM schemes: Give your CUSUM a head start. Technometrics, 24(3), 199- 205.
[16] Montgomery, D.C. (1985). Introduction to Statistical Quality Control. John Wiley & Sons, New York.
[17] Mood, A.M., Graybill, F.A. , & Boes, D.C. (1974). Introduction to the Theory of Statistics. McGraw-Hill, Tokyo.
[18] Mukherjee, A., Graham, M. A., Chakraborti S. (2013). Distribution-free exceedance CUSUM control charts for location. Communications in Statistics – Simulation and Computation, 42(5), 1153– 1187.
[19] Polunchenko, A. S. (2016). A Note on Efficient Performance Evaluation of the Cumulative Sum Chart and the Sequential Probability Ratio Test. Applied Stochastic Models in Business and Industry, Accepted.
[20] Rohatgi, V.K. , & Ehsanes Saleh, A.K. (2001). An Introduction to Probability and Statistics. John Wiley & Sons, New York, 2nd edition.
[21] Srinivasa Rao, G. (2013). One-sided cumulative sum (CUSUM) control charts for the erlang-truncated exponential distribution. Computational Methods in Science and Technology, 19(4), 229-234.
[22] Wieringa, J.E. (1999). Statistical process control for serially correlated data. PhD Thesis. University of Groningen, Netherlands.
[23] Zanella, A. (1991). On the relation between Wald’s sequential tests and the cusum control charts for sample means: correcting a wrong interpretation. Statistica Applicata, 4, 493-519.