Abstract
The most popular multivariate process monitoring and control procedure used in the industry is the chi-square control chart. As with most Shewhart-type control charts, the major disadvantage of the chi-square control chart, is that it only uses the information contained in the most recently inspected sample; as a consequence, it is not very efficient in detecting gradual or small shifts in the process mean vector. During the last decades, the performance improvement of the chi-square control chart has attracted continuous research interest. In this paper we introduce a simple modification of the chi-square control chart which makes use of the notion of runs to improve the sensitivity of the chart in the case of small and moderate process mean vector shifts.
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S. Aki, “Waiting time problems for a sequence of discrete random variables,” Annals of the Institute of Statistical Mathematics vol. 44 pp. 363–378, 1992.
D. L. Antzoulakos, “Waiting times for patterns in a sequence of multistate trials,” Journal of Applied Probabilty vol. 38 pp. 508–518, 2001.
F. B. Alt, and N. D. Smith, “Multivariate process control.” In P. R. Krishnaiah and C. R. Rao (eds.), Handbook of Statistics vol. 7 pp. 333–351, Elsevier: Amsterdam, The Netherlands, 1988.
L. C. Alwan, “Cusum quality control–Multivariate approach,” Communications in Statistics– Theory in Methods vol. 15 pp. 3531–3543, 1986.
F. Aparisi, C. W. Champ, and J. C. G. Diaz, “A performance analysis of Hotelling’s χ 2 control chart with supplementary runs rules,” Quality Engineer vol. 16 pp. 13–22, 2004.
N. Balakrishnan, and M. V. Koutras, Runs and Scans with Applications: Wiley: New York, 2002.
C. W. Champ, and W. H. Woodall, “Exact results for Shewhart control charts with supplementary runs rules,” Technometrics vol. 29 pp. 293–399, 1987.
R. B. Crosier, “Multivariate generalizations of cumulative sum quality-control schemes,” Technometrics vol. 30 pp. 291–303, 1988.
J. C. Fu, and Y. M. Chang, “On ordered series and later waiting time distributions in a sequence of Markov dependent multistate trials,” Journal of Applied Probabilty vol. 40(3) pp. 623–642, 2003.
J. C. Fu, and W. Y. W. Lou, Distribution Theory of Runs and Patterns and Its Applications, World Scientific: New Jersey, 2003.
J. C. Fu, F. A. Spiring, and H. S. Xie, “On the average run lengths of quality control schemes using a Markov chain approach,” Statistics and Probability Letters vol. 56(4) pp. 369–380, 2002.
J. C. Fu, G. Shmueli, and Y. M. Chang, “A unified Markov chain approach for computing the run length distribution in control charts with simple or compound rules,” Statistics and Probability Letters vol. 65 pp. 457–466, 2003.
D. M. Hawkins, “Multivariate quality control based on regression-adjusted variables,” Technometrics vol. 33 pp. 61–75, 1991.
H. Hotelling, “Multivariate quality control, illustrated by the air testing of sample bombsights,” In C. Eisenhart, M. W. Hastay, and W. A. Wallis (eds.), Techniques of Statistical Analysis, pp. 111–184, McGraw-Hill: New York, 1947.
N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, Wiley: New York, vol. 1, 1994.
N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, Wiley: New York, vol. 2, 1995.
M. B. C. Khoo, and S. H. Quah, “Incorporating runs rules into Hotelling’s χ 2 control charts,” Quality Engineer vol. 15 pp. 671–675, 2003.
M. V. Koutras, “Waiting time distributions associated with runs of fixed length in two-state Markov chains,” Annals of the Institute of Statistical Mathematics vol. 49 pp. 123–139, 1997.
C. A. Lowry, and D. C. Montgomery, “A review of multivariate control charts,” IIE Transactions vol. 27 pp. 800–810, 1995.
D. C. Montgomery, Introduction to Statistical Process Control, Wiley: New York, 2001.
E. S. Page, “Control charts with warning lines,” Biometrics vol. 42 pp. 243–257, 1955.
J. J. Pignatiello Jr., and G. C. Runger, “Comparisons of multivariate CUSUM charts” Journal of Quality and Technology vol. 22 pp. 173–186, 1990.
S. E. Ridgon, “A double-integral equation for the average run length of multivariate exponentially weighted moving average control chart,” Statistics and Probability Letters vol. 24 pp. 365–373, 1995.
T. P. Ryan, Statistical Methods for Quality Improvement, Wiley: New York, 2000.
G. Shmueli, “System-Wide probabilities for systems with runs and scans rules,” Methodology and Computing in Applied Probability vol. 4 pp. 401–419, 2003.
G. Shmueli, and A. Cohen, “Run-related probability functions applied to sampling inspection,” Technometrics vol. 42(2) pp. 188–202, 2000.
G. Shmueli, and A. Cohen, “Run-length distribution for control charts with runs and scans rules,” Communication in Statistics-Theory and Methods vol. 32 pp. 475–495, 2003.
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Koutras, M.V., Bersimis, S. & Antzoulakos, D.L. Improving the Performance of the Chi-square Control Chart via Runs Rules. Methodol Comput Appl Probab 8, 409–426 (2006). https://doi.org/10.1007/s11009-006-9754-z
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DOI: https://doi.org/10.1007/s11009-006-9754-z