Abstract
Misleading signals (MS) correspond to the misinterpretation of a shift in the process mean (variance) as a shift in the process variance (mean). MS occur when:
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The individual chart for the mean triggers a signal before the one for the variance, even though the process mean is on-target and the variance is off-target;
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The individual chart for the variance triggers a signal before the one for the mean, although the variance is in-control and the process mean is out-of-control.
MS can lead to a misdiagnosis of assignable causes and to incorrect actions to bring the process back to target. Unsurprisingly, the performance assessment of simultaneous schemes for the process mean and variance requires not only the use of run length (RL) related performance measures, but also the probability of misleading signals (PMS). We assess the impact of autocorrelation on the PMS of simultaneous Shewhart and EWMA residual schemes for the mean and variance of stationary AR(1), AR(2) and ARMA(1,1) processes. This assessment is done by means of some stochastic ordering results and some illustrations.
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References
Antunes, C. (2009). Avaliação do impacto da correlação em sinais erróneos de esquemas de conjuntos para o valor esperado e variância (Assessment of the impact of the correlation on misleading signals in joint schemes for the mean and variance). Master’s thesis, Instituto Superior Técnico, Technical University of Lisbon.
Brockwell, P. J., & Davis, R. A. (1991). Time series: Theory and methods. New York: Springer.
Brook, D., & Evans, D. A. (1972). An approach to the probability distribution of CUSUM run length. Biometrika,59, 539–549.
St. John, R. C., & Bragg, D. J. (1991). Joint X-bar R charts under shift in mu or sigma. ASQC Quality Congress Transactions – Milwaukee,45, 547–550.
Knoth, S., & Schmid, W. (2002). Monitoring the mean and the variance of a stationary process. Statistica Neerlandica,56, 77–100.
Knoth, S., Morais, M. C., Pacheco, A., & Schmid, W. (2009). Misleading signals in simultaneous residual schemes for the mean and variance of a stationary process. Communications in Statistics – Theory and Methods,38, 2923–2943.
Mathai, A. M., & Provost, S. B. (1992). Quadratic forms in random variables. New York: Marcel Dekker.
Morais, M. J. C. (2002). Stochastic ordering in the performance analysis of quality control schemes. Ph.D. thesis, Instituto Superior Técnico, Technical University of Lisbon.
Morais, M. C., & Pacheco, A. (1998). Two stochastic properties of one-sided exponentially weighted moving average control charts. Communications in Statistics – Simulation and Computation,27, 937–952.
Morais, M. C., & Pacheco, A. (2000). On the performance of combined EWMA schemes for μ and σ: A Markovian approach. Communications in Statistics – Simulation and Computation,29, 153–174.
Morais, M. C., & Pacheco, A. (2006). Misleading signals in joint schemes for μ and σ. In H. J. Lenz & P. T. Wilrich (Eds.), Frontiers in statistical quality control (Vol. 8, pp. 100–102). Heidelberg: Physica-Verlag.
Ramos, P. F., Morais, M. C., & Pacheco, A. (2010). Misleading signals in simultaneous residual schemes for the process mean and variance of AR(1) processes: A stochastic ordering approach (submited for publication).
Reynolds, M. R., Jr., & Stoumbos, Z. G. (2001). Monitoring the process mean and variance using individual observations and variable sampling intervals. Journal of Quality Technology,33, 181–205.
Reynolds, M. R., Jr., & Stoumbos, Z. G. (2004). Control charts and the efficient allocation of sampling resources. Technometrics,46, 200–214.
Acknowledgements
The first author is supported by grant SFRH/BD/35739/2007 of Fundação para a Ciência e a Tecnologia (FCT) and was partially supported by Centro de Matemática e Aplicações (CEMAT) and FCT while visiting the Department of Statistics of the European University Viadrina (Frankfurt (Oder), Germany).
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Ramos, P.F., Morais, M.C., Pacheco, A., Schmid, W. (2012). Assessing the Impact of Autocorrelation in Misleading Signals in Simultaneous Residual Schemes for the Process Mean and Variance: A Stochastic Ordering Approach. In: Lenz, HJ., Schmid, W., Wilrich, PT. (eds) Frontiers in Statistical Quality Control 10. Frontiers in Statistical Quality Control, vol 10. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-2846-7_3
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DOI: https://doi.org/10.1007/978-3-7908-2846-7_3
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