Journal of Korean Society for Quality Management (품질경영학회지)

Korean Society for Quality Management (한국품질경영학회)
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Cusum Control Chart for Monitoring Process Variance

공정분산 관리를 위한 누적합 관리도

  • Lee, Yoon-Dong (Department of Applied Statistics, Konkuk University) ;
  • Kim, Sang-Ik (Department of Applied Statistics, Konkuk University)
  • 이윤동 (건국대학교 응용통계학과) ;
  • 김상익 (건국대학교 응용통계학과)
  • Published : 2005.09.30
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Abstract

Cusum control chart is used for the purpose of controling the process mean. We consider the problem related to cusum chart for controling process variance. Previous researches have considered the same problem. The main difficulty shown in the related researches was to derive the ARL function which characterizes the properties of the chart. Sample variance, differently with sample mean, follows chi-squared type distribution, even when the quality characteristics are assumed to be normally distributed. The ARL function of cusum is described by a type of integral equation. Since the solution of the integral equation for non-normal distribution is not known well, people used simulation method instead of solving the integral equation directly, or approximation method by taking logarithm of the sample variance. Recently a new method to solve the integral equation for Erlang distribution was published. Here we consider the steps to apply the solution to the problem of controling process variance.

Keywords

References

  1. 이은경, 나명환, 이윤동(2005), '얼랑분포의 축자확률비 검정과 관련된 적분방정식의 해', '응용통계연구', 18권, pp.57-66 https://doi.org/10.5351/KJAS.2005.18.1.057
  2. Brook, D. and Evans, D. A.(1972). 'An Approach to the Probability Distributionb of Cusum Run Length', Biometriks, Vol. 59, pp. 539-549 https://doi.org/10.1093/biomet/59.3.539
  3. Chang, T. and Gan, F.(1995), 'A Cumulative Sum Control Chart for Monitoring Process Variance', Journal of Quality Technology, Vol. 25, pp. 109-119
  4. Crowder, S. V. and Hamilton, M. D.(1992), 'An EWMA for Monitoring a Process Standard Deviation', Journal of Quality Technology, Vol. 24, .pp 12-21 https://doi.org/10.1080/00224065.1992.11979369
  5. Gan, F. and Choi, K(1994), 'Computing Average Run Lengths for Exponential CUSUM Schemes',Journal of Quality Technology, Vol. 26,pp 134-139 https://doi.org/10.1080/00224065.1994.11979513
  6. Knoth, S.(1998), 'Exact Average Run Leng ths of CUSUM Schemes for Erlang Distribution' Sequentl Analysis, Vol. 17, pp. 173-184 https://doi.org/10.1080/07474949808836405
  7. Kohlruss, D.(1994), 'Exact Formulas for the OC and the ASN Functions of the SPRT for Erlang Fistributions', Sequential Analysis, Vol. 13, pp. 53-62 https://doi.org/10.1080/07474949408836293
  8. Lee, Y. D.(2004), 'Unified Solutions of Integral Equations of SPRT for exponential Random Variables', Commanicetions in Statistics, Series A, Theory and Method, Vol. 33, pp. 65-74
  9. Page, E. S.(1954), 'Continuous Inspection Schemes', Biometrika, Vol. 41, pp. 100-115 https://doi.org/10.1093/biomet/41.1-2.100
  10. Stadje, W.(1987). 'On the SPRT for the Mean of an Exponential Distribution', Statistics & Probability Letters, Vol. 5, pp. 389-395 https://doi.org/10.1016/0167-7152(87)90088-5
  11. Vardeman, S. and Ray, D.(1985), 'Average Run Lengths for CUSUM Schemes When Observations Are Exponentially Distributed', Technometrics, Vol. 27, pp. 145-150 https://doi.org/10.2307/1268762