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Misleading Signals in Simultaneous Schemes for the Mean Vector and Covariance Matrix of a Bivariate Process

  • Patrícia Ferreira Ramos
  • Manuel Cabral Morais
  • António Pacheco
  • Wolfgang Schmid
Chapter
Part of the Studies in Theoretical and Applied Statistics book series (STAS)

Abstract

In a bivariate setting, misleading signals (MS) correspond to valid alarms which lead to the misinterpretation of a shift in the mean vector (resp. covariance matrix) as a shift in the covariance matrix (resp. mean vector). While dealing with bivariate output and two univariate control statistics (one for each parameter), MS occur when:
  • The individual chart for the mean vector triggers a signal before the one for the covariance matrix, although the mean vector is on-target and the covariance matrix is off-target.

  • The individual chart for the variance triggers a signal before the one for the mean, despite the fact that the covariance matrix is in-control and the mean vector is out-of-control.

Since MS can be rather frequent in the univariate setting, as reported by many authors, this chapter thoroughly investigates the phenomenon of MS in the bivariate case.

Keywords

Covariance Matrix Control Chart Bivariate Normal Distribution Absorb Markov Chain Individual Chart 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The authors thank the financial support of Centro de Matemática e Aplicações (CEMAT) and Fundação para a Ciência e a Tecnologia (FCT). The first author was also supported by grant SFRH/BD/35739/2007 of FCT and would like to thank all the members of the Department of Statistics of the European University Viadrina (Frankfurt Oder, Germany) for their hospitality.

References

  1. 1.
    Alt, F.: Multivariate quality control. In: Kotz, S., Johnson, N.L, Read, C.R. (eds.) The Encyclopedia of Statistical Sciences, pp. 110–122. Wiley, New York (1984)Google Scholar
  2. 2.
    Antunes, C.: Avaliação do impacto da correlação em sinais erróneos em esquemas 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). Instituto Superior Técnico, Universidade Técnica de Lisboa (2009)Google Scholar
  3. 3.
    Brook, D., Evans, D.A.: An approach to the probability distribution of CUSUM run length. Biometrika 59, 539–549 (1972)Google Scholar
  4. 4.
    Chen, G., Cheng, S.W., Xie, H.: A new multivariate control chart for monitoring both location and dispersion. Comm. Stat. Simul. Comput. 34, 203–217 (2005)Google Scholar
  5. 5.
    Hotelling, H.: Multivariate quality control illustrated by the air testing of sample bombsights. In: Eisenhart, C., Hastay, M.W., Wallis, W.A. (eds.) Techniques of Statistical Analysis, pp. 111–184. McGraw Hill, New York (1947)Google Scholar
  6. 6.
    Karr, A.F.: Probability. Springer, New York (1993)Google Scholar
  7. 7.
    Khoo, B.C.: A new bivariate control chart to monitor the multivariate process mean and variance simultaneously. Qual. Eng. 17, 109–118 (2005)Google Scholar
  8. 8.
    Knoth, S., Morais, M.C., Pacheco, A., Schmid, W.: Misleading signals in simultaneous residual schemes for the mean and variance of a stationary process. Comm. Stat. Theory Methods 38, 2923–2943 (2009)Google Scholar
  9. 9.
    Lowry, C.A., Montgomery, D.C.: A review of multivariate control charts. IIE Trans. 27, 800–810 (1995)Google Scholar
  10. 10.
    Machado, M.A.G., Costa, A.F.B.: Monitoring the mean vector and the covariance matrix of bivariate processes. Brazilian J. Oper. Produc. Manag. 5, 47–62 (2008)Google Scholar
  11. 11.
    Mathai, A.M., Provost, S.B.: Quadratic Forms in Random Variables. Marcel Dekker, New York (1992)Google Scholar
  12. 12.
    Morais, M.C., Okhrin, Y., Pacheco, A., Schmid, W.: EWMA charts for multivariate output: some stochastic ordering results. Comm. Stat. Theory Methods 37, 2653–2663 (2008)Google Scholar
  13. 13.
    Morais, M.C., Pacheco, A.: On the performance of combined EWMA schemes for μ and σ: a Markovian approach. Commun. Stat. Simul. Comput. 29, 153–174 (2000)Google Scholar
  14. 14.
    Morais, M.C., Pacheco, A.: Misleading signals in joint schemes for μ and σ. In: Lenz, H.J., Wilrich, P.T. (eds.) Frontiers in Statistical Quality Control, vol. 16, pp. 100–122. Physica-Verlag, Heidelberg (2006)Google Scholar
  15. 15.
    Morais, M.J.C.: Stochastic ordering in the performance analysis of quality control schemes. Ph.D. thesis, Instituto Superior Técnico, Technical University of Lisbon (2002)Google Scholar
  16. 16.
    Ramos, P.F., Morais, M.C., Pacheco, A.: Misleading signals in simultaneous residual schemes for the process mean and variance of AR(1) processes: a stochastic ordering approach (accepted for publication in the international book series “Studies in Theoretical and Applied Statistics”) (2012)Google Scholar
  17. 17.
    Ramos, P.F., Morais, M.C., Pacheco, A., Schmid, W.: Assessing the impact of autocorrelation in misleading signals in simultaneous residual schemes for the process mean and variance: a stochastic ordering approach (accepted for publication in “Frontiers on Statistical Quality Control 10”) (2011)Google Scholar
  18. 18.
    Reynolds, M.R. Jr, Stoumbos, Z.G.: Monitoring the process mean and variance using individual observations and variable sampling intervals. J. Qual. Technol. 33, 181–205 (2001)Google Scholar
  19. 19.
    Reynolds, M.R. Jr, Stoumbos, Z.G.: Control charts and the efficient allocation of sampling resources. Technometrics 46, 200–214 (2004)Google Scholar
  20. 20.
    St. John, R.C., Bragg, D.J.: Joint X-bar R charts under shift in mu or sigma. ASQC Quality Congress Transactions — Milwaukee, pp. 547–550 (1991)Google Scholar
  21. 21.
    Tang, P.F., Barnett, N.S.: Dispersion control for multivariate processes. Aust. J. Stat. 38(3), 235–251 (1996)Google Scholar
  22. 22.
    Tang, P.F., Barnett, N.S.: Dispersion control for multivariate processes – some comparisons. Aust. J. Stat. 38(3), 253–273 (1996)Google Scholar
  23. 23.
    Tong, Y.L.: The Multivariate Normal Distribution. Springer, New York (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Patrícia Ferreira Ramos
    • 1
  • Manuel Cabral Morais
    • 2
  • António Pacheco
    • 2
  • Wolfgang Schmid
    • 3
  1. 1.CEMAT, Instituto Superior TécnicoTechnical University of LisbonLisboaPortugal
  2. 2.CEMAT and Mathematics Department, Instituto Superior TécnicoTechnical University of LisbonLisboaPortugal
  3. 3.Department of StatisticsEuropean University ViadrinaFrankfurt (Oder)Germany

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