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A Simple Relation between the Pattern Probability and the Rate of False Signals in Control Charts

Published online by Cambridge University Press:  27 July 2009

Frank K. Hwang  and
Dan Trietsch
Frank K. Hwang
Affiliation:
AT&T Bell Laboratories, Murray Hill, New Jersey 07974
Dan Trietsch
Affiliation:
Trietsch Associates, 2804 Forest Hill Boulevard, Pacific Grove, California 93950
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Abstract

A special pattern of consecutive points on a control chart often suggests that the process is out of control due to a special cause. Usually, the probability of such a pattern is not difficult to calculate, but its relation to the rate of a false signal in a control chart is unclear. In this paper we give such a relation for a wide class of patterns that includes six of the eight patterns of special cause discussed in the AT&T Statistical Quality Control Handbook (1958). We derive a surprisingly simple formula for the rate of a false signal in terms of the pattern probability and use this result to improve on the test proposed in the handbook. We also use this formula to give a simple proof of a result of Wolfowitz (1944, Annals of Mathematical Statistics 15: 163–172) and Olmstead (1945, Annals of Mathematical Statistics 17: 24–33) on the monotone sequence. Finally, we give an approximation of the rate of a false signal for another wide class of patterns that covers the two remaining patterns of special cause in the AT&T handbook.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 10 , Issue 3 , July 1996 , pp. 315 - 323
Copyright
Copyright © Cambridge University Press 1996

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References

1. AMS Problem 4755. (1958). American Mathematical Monthly 67: 533–534.Google Scholar
2. AT&T. (1958). Statistical quality control handbook, 2nd edition. Delmar, NC: AT&T.Google Scholar
3.Bissell, A.F. (1978). An attempt to unify the theory of quality control procedures. Bulletin in Applied Statistics 5: 113–128.CrossRefGoogle Scholar
4.Godbole, A.P. (1991). Poisson approximations for runs and patterns of rare events. Advances in Applied Probabilities 23: 851–865.CrossRefGoogle Scholar
5.Greenberg, I. (1970). The first occurrence of n successes in N trials. Technometrics 12: 627–634.CrossRefGoogle Scholar
6.Nelson, L.S. (1984). The Shewhart control chart-Tests for special causes. Journal of Quality Control 16: 237–239.Google Scholar
7.Nelson, L.S. (1985). Interpreting Shewhart control charts. Journal of Quality Control 17: 114–116.Google Scholar
8.Olmstead, P.S. (1945). Distribution of simple arrangements for runs up and down. Annals of Mathematical Statistics 17: 24–33.CrossRefGoogle Scholar
9.Page, E.S. (1955). Control charts with warning lines. Biometrika 42: 243–257.CrossRefGoogle Scholar
10.Roberts, S.W. (1958). Properties of control chart zone tests. Bell System Technical Journal 37: 83–114.Google Scholar
11.Saperstein, B. (1973). On the occurrence of n successes within N Bernoulli trials. Technometrics 15: 809–818.Google Scholar
12.Wolfowitz, J. (1944). Asymptotic distributions of runs up and down. Annals of Mathematical Statistics 15: 163–172.CrossRefGoogle Scholar