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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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