Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-04-30T23:56:41.662Z Has data issue: false hasContentIssue false

On a unified approach to the analysis of two-sided cumulative sum control schemes with headstarts

Published online by Cambridge University Press:  01 July 2016

Emmanuel Yashchin*
Affiliation:
IBM Corporation
*
Postal address: IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA.

Abstract

The necessary and sufficient conditions for various modes of interactions of the upper and lower schemes with headstarts are derived. The expression for the Laplace transform of the run-length distribution (for both interacting and non-interacting schemes) is obtained and used to develop a method of analysis for general two-sided cumulative sum schemes with headstarts. The results are shown to be relevant in the case when the schemes are supplemented by Shewhart’s control limits.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1985 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bagshaw, M. and Johnson, R. A. (1975) The effect of serial correlation on the performance of CUSUM tests II. Technometrics 17, 7380.Google Scholar
Barnard, G. (1959) Control charts and stochastic processes. J. R. Statist. Soc. B 21, 239257.Google Scholar
Bissell, A. (1969) Cusum techniques for quality control. Appl. Statist. 18, 130.CrossRefGoogle Scholar
Brook, D. and Evans, D. A. (1972) An approach to the probability distribution of cusum run length. Biometrika 59, 539549.CrossRefGoogle Scholar
Van Dobben De Bruyn, D. S. (1968) Cumulative Sum Tests: Theory and Practice. Hafner, New York.Google Scholar
Gantmacher, F. R. (1964) The Theory of Matrices , Vols 1, 2. Chelsea, New York.Google Scholar
Goel, A. L. and Wu, S. M. (1971) Determination of ARL and a contour nomogram for cusum charts to control normal mean. Technometrics 13, 221230.Google Scholar
Gradsteyn, I. S. and Ryzhik, I. M. (1980) Table of Integrals, Series, and Products. Academic Press, New York.Google Scholar
Johnson, R. A. and Bagshaw, M. (1974) The effect of serial correlation on the performance of CUSUM tests. Technometrics 16, 103122.CrossRefGoogle Scholar
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
Kemp, K. (1958) Formulae for calculating the operating characteristics and the ASN of some sequential tests. J. R. Statist. Soc. B 20, 379386.Google Scholar
Kemp, K. (1961) The average run length of the cumulative sum chart when a V-mask is used. J. R. Statist. Soc. B 23, 149153.Google Scholar
Khan, R. A. (1981) A note on Page’s two-sided cumulative sum procedure. Biometrika 68, 717719.Google Scholar
Lucas, J. M. and Crosier, R. B. (1982a) Fast initial response for cusum quality control schemes: Give your cusum a head start. Technometrics 24, 199205.Google Scholar
Lucas, J. M. and Crosier, R. B. (1982b) Robust cusum: A robustness study for cusum quality control schemes. Commun. Statist. Theory Methods A 11, 26692687.Google Scholar
Nadler, J. and Robbins, N. B. (1971) Some characteristics on Page’s two-sided procedure for detecting a change in a location parameter. Ann. Math. Statist. 42, 538551.Google Scholar
Page, E. (1954) Continuous inspection schemes. Biometrika 41, 100115.Google Scholar
Reynolds, M. (1975) Approximations to the average run length in cumulative sum control charts. Technometrics 17, 6571.CrossRefGoogle Scholar
Woodall, W. H. (1983) The distribution of the run length of one-sided cusum procedures for continuous random variables. Technometrics 25, 295300.Google Scholar
Woodall, W. H. (1984) On the Markov chain approach to the two-sided CUSUM procedure. Technometrics 26, 4146.Google Scholar
Zacks, S. (1981) The probability distribution and the expected value of a stopping variable associated with one-sided cusum procedures for non-negative integer valued random variables. Commun. Statist. Theory Methods , A10, 22452258.Google Scholar