IDEAS home Printed from https://ideas.repec.org/a/bpj/strimo/v21y2003i1-2003p79-92n8.html
   My bibliography  Save this article

Tail behaviour of a general family of control charts

Author

Listed:
  • Schmid Wolfgang
  • Okhrin Yarema

Abstract

In this paper we consider a general control scheme. The control statistic Zt is equal to an arbitrary weighted sum of the past observations Xt,...,X1. This approach covers most of the applied control schemes like for instance moving average, EWMA and ARMA(1,1) charts. The process {Xt} is assumed to be a stationary Gaussian process. The aim of the work is to analyze the behaviour of the tail probability of the run length N=inf{t∈ℕ:Zt−E(Zt)>c√{Var(Zt)}} with respect to the autocorrelation of {Xt}. It is shown under which conditions on the weights and on the autocorrelations of {Xt} the correlation between Zt and Zt−i is a nondecreasing function in the autocorrelations of the observed process. Using this result it can be proved that the probability of a false alarm is a nondecreasing function of the autocorrelations of {Xt}, too. The weight conditions are verified for several well-known charts.

Suggested Citation

  • Schmid Wolfgang & Okhrin Yarema, 2003. "Tail behaviour of a general family of control charts," Statistics & Risk Modeling, De Gruyter, vol. 21(1/2003), pages 79-92, January.
    • Handle: RePEc:bpj:strimo:v:21:y:2003:i:1/2003:p:79-92:n:8
    DOI: 10.1524/stnd.21.1.79.20320
    as

    Download full text from publisher

    File URL: https://doi.org/10.1524/stnd.21.1.79.20320
    Download Restriction: For access to full text, subscription to the journal or payment for the individual article is required.
    File URL: https://libkey.io/10.1524/stnd.21.1.79.20320?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Don G. Wardell & Herbert Moskowitz & Robert D. Plante, 1992. "Control Charts in the Presence of Data Correlation," Management Science, INFORMS, vol. 38(8), pages 1084-1105, August.
    2. M. Pawlak & W. Schmid, 2001. "On the Distributional Properties of GARCH Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 22(3), pages 339-352, May.
    3. Alwan, Layth C & Roberts, Harry V, 1988. "Time-Series Modeling for Statistical Process Control," Journal of Business & Economic Statistics, American Statistical Association, vol. 6(1), pages 87-95, January.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Marta Benková & Dagmar Bednárová & Gabriela Bogdanovská & Marcela Pavlíčková, 2023. "Use of Statistical Process Control for Coking Time Monitoring," Mathematics, MDPI, vol. 11(16), pages 1-30, August.
    2. Ord, J. Keith & Koehler, Anne B. & Snyder, Ralph D. & Hyndman, Rob J., 2009. "Monitoring processes with changing variances," International Journal of Forecasting, Elsevier, vol. 25(3), pages 518-525, July.
    3. Ridley, D. & Duke, D., 2007. "Moving -window spectral model based statistical process control," International Journal of Production Economics, Elsevier, vol. 105(2), pages 492-509, February.
    4. Gulser Koksal & Burcu Kantar & Taylan Ali Ula & Murat Caner Testik, 2008. "The effect of Phase I sample size on the run length performance of control charts for autocorrelated data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 35(1), pages 67-87.
    5. Samari, Goleen & Catalano, Ralph & Alcalá, Héctor E. & Gemmill, Alison, 2020. "The Muslim Ban and preterm birth: Analysis of U.S. vital statistics data from 2009 to 2018," Social Science & Medicine, Elsevier, vol. 265(C).
    6. Amira Dridi & Mohamed El Ghourabi & Mohamed Limam, 2012. "On monitoring financial stress index with extreme value theory," Quantitative Finance, Taylor & Francis Journals, vol. 12(3), pages 329-339, March.
    7. Weihs, Claus & Theis, Winfried & Messaoud, Amor & Hering, Franz, 2004. "Monitoring of the BTA Deep Hole Drilling Process Using Residual Control Charts," Technical Reports 2004,60, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    8. Fernandes, Marcelo, 2004. "Bounds for the probability distribution function of the linear ACD process," Statistics & Probability Letters, Elsevier, vol. 68(2), pages 169-176, June.
    9. Johannes Freiesleben & Nicolas Gu'erin, 2015. "Homogenization and Clustering as a Non-Statistical Methodology to Assess Multi-Parametrical Chain Problems," Papers 1505.03874, arXiv.org, revised Dec 2017.
    10. Miguel Flores & Salvador Naya & Rubén Fernández-Casal & Sonia Zaragoza & Paula Raña & Javier Tarrío-Saavedra, 2020. "Constructing a Control Chart Using Functional Data," Mathematics, MDPI, vol. 8(1), pages 1-26, January.
    11. Timothy M. Young & Ampalavanar Nanthakumar & Hari Nanthakumar, 2021. "On the Use of Copula for Quality Control Based on an AR(1) Model," Mathematics, MDPI, vol. 9(18), pages 1-13, September.
    12. Thaga K. & Kgosi P. M. & Gabaitiri L., 2007. "Max-Chart for Autocorrelated Processes," Stochastics and Quality Control, De Gruyter, vol. 22(1), pages 87-105, January.
    13. Ramjee, Radhika & Crato, Nuno & Ray, Bonnie K., 2002. "A note on moving average forecasts of long memory processes with an application to quality control," International Journal of Forecasting, Elsevier, vol. 18(2), pages 291-297.
    14. P. Vellaisamy & S. Sankar & M. Taniguchi, 2003. "Estimation and Design of Sampling Plans for Monitoring Dependent Production Processes," Methodology and Computing in Applied Probability, Springer, vol. 5(1), pages 85-108, March.
    15. A. Snoussi, 2011. "SPC for short-run multivariate autocorrelated processes," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(10), pages 2303-2312.
    16. Messaoud, Amor & Weihs, Claus & Hering, Franz, 2008. "Detection of chatter vibration in a drilling process using multivariate control charts," Computational Statistics & Data Analysis, Elsevier, vol. 52(6), pages 3208-3219, February.
    17. Žmuk Berislav, 2016. "Capabilities of Statistical Residual-Based Control Charts in Short- and Long-Term Stock Trading," Naše gospodarstvo/Our economy, Sciendo, vol. 62(1), pages 12-26, March.
    18. Mohamed El Ghourabi & Amira Dridi & Mohamed Limam, 2015. "A new financial stress index model based on support vector regression and control chart," Journal of Applied Statistics, Taylor & Francis Journals, vol. 42(4), pages 775-788, April.
    19. Esmeralda Gonçalves & Joana Leite & Nazaré Mendes-Lopes, 2013. "The ARL of modified Shewhart control charts for conditionally heteroskedastic models," Statistical Papers, Springer, vol. 54(1), pages 1-19, February.
    20. Hwarng, H. Brian, 2001. "Insights into neural-network forecasting of time series corresponding to ARMA(p,q) structures," Omega, Elsevier, vol. 29(3), pages 273-289, June.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bpj:strimo:v:21:y:2003:i:1/2003:p:79-92:n:8. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Peter Golla (email available below). General contact details of provider: https://www.degruyter.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.