Abstract
In this paper we discuss the behavior of the Shewhart residual chart and the modified Shewhart chart if the parameters of the underlying process are unknown and thus have to be estimated. We focus on the estimation of the variance. For AR models we also consider the estimation of the AR coefficients. The average run length (ARL) of the control chart with estimated parameters is compared with the ARL of the scheme for known parameters and with the ARL for independent variables. Additionally, we give recommendations on the choice of the estimators in the context of Shewhart control schemes.
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References
Alwan, L. C. and Roberts, H. V. (1988). Time-series modeling for statistical process control.Journal of Business and Economic Statistics, 6(1), 87–95.
Box, G. E. P., Jenkins, G. M. and Reinsel, G. C. (1994).Time Series Analysis-Forecasting and Control, Prentice Hall, Englewood Cliffs.
Brockwell, P. J. and Davis, R. A. (1991).Time Series: Theory and Methods. Springer, New York.
Brook, D. and Evans, D. A. (1972). An approach to the probability distributions of CUSUM run length.Biometrika, 59, 3, 539–549.
Harris, T. J. and Ross, W. H. (1991). Statistical process control procedures for correlated observations.Canadian Journal of Chemical Engineering, 69, 48–57.
Kramer, H. G. (1997).On Control Charts for Time Series, Ph.D. thesis, University of Uhn, Germany.
Kramer, H. G. and Schmid, W. (1997). EWMA charts for multivariate time series.Sequential Analysis, 16, 131–154.
Lucas, J. M. and Saccucci, M. S. (1990). Exponentially weighted moving average control schemes: properties and enhancements.Technometrics, 32(1), 1–29.
Maragah, H. D. and Woodall, W. D. (1992). The effect of autocorrelation on the retrospective X-chart.Journal of Statist. Comput. Simul., 40, 29–42.
Montgomery, D. C. and Mastrangelo, C. M. (1991). Some statistical process control methods for autocorrelated data.Journal of Quality Technology, 23(3), 179–204.
Notohardjono, D. and Ermer, D. S. (1986). Time series control charts for correlated and contaminated data.Journal of Engineering for Industry, 108, 219–226.
Ryan, T. P. (1991). Discussion of “Some statistical process control methods for autocorrelated data”.Journal of Quality Technology, 23(3), 200–202.
Schmid, W. (1995). On the run length of a Shewhart chart for correlated data.Statistical Papers, 36, 111–130.
Schmid, W. (1997a). On EWMA charts for time series, In:Frontiers in Statistical Quality Control 5, Lenz, H.-J. and Wilrich, P.-Th. (Eds.). Physica-Verlag, Heidelberg, Germany, 115–137.
Schmid, W. (1997b). CUSUM control schemes for Gaussian processes.Statistical Papers, 38, 191–217.
Severin, T. and Schmid, W. (1996). Monitoring changes in GARCH models. Technical Report No. 64, Europe-University Viadrina, Frankfurt (Oder), Germany.
Taniguchi, M. (1991).Higher Order Asymptotic Theory for Time Series Analysis. Lecture Notes in Statistics, Vol. 68, Springer, New York.
Tong, Y. L. (1980).Probability Inequalities in Multivariate Distributions. Academic Press, New York.
Tong, Y. L. (1990).The Multivariate Normal Distribution. Springer, New York.
Tseng, S. and Adams, B. M. (1994). Monitoring autocorrelated processes with an exponentially weighted moving average forecast.Journal of Statist. Comput. Simul., 50, 187–195.
Vander Wiel, S. A. (1996). Monitoring processes that wander using integrated moving average models.Technometrics, 38(2), 139–151.
Vasilopoulos, A. V. and Stamboulis, A. P. (1978). Modification of control chart limits in the presence of data correlation.Journal of Quality Technology, 10(1), 20–30.
Wardell, D. G., Moskowitz, H. and Plante, R. D. (1992). Control charts in the presence of data correlation.Management Science, 38(8), 1084–1105.
Wardell, D. G., Moskowitz, H. and Plante, R. D. (1994). Run-length distributions of special-cause control charts for correlated processes.Technometrics, 36(1), 3–27.
Yashchin, E. (1993). Performance of CUSUM control schemes for serially correlated observations.Technometrics, 35(1), 35–52.
Holger Kramer and Wolfgang Schmid, Europe University Frankfurt (Oder), Department of Statistics, P.O. Box 776, D-15207 Frankfurt (Oder), Germany.
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Kramer, H., Schmid, W. The influence of parameter estimation on the ARL of Shewhart type charts for time series. Statistical Papers 41, 173–196 (2000). https://doi.org/10.1007/BF02926102
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DOI: https://doi.org/10.1007/BF02926102