116
Views
5
CrossRef citations to date
0
Altmetric
Original Article

Explicit analytical solutions for ARL of CUSUM chart for a long-memory SARFIMA model

, ORCID Icon & ORCID Icon
Pages 1176-1190 | Received 17 Oct 2016, Accepted 20 Nov 2017, Published online: 18 Dec 2017

References

  • Abujiya, M. R., M. Riaz, and M. H. Lee. 2015. Enhanced cumulative sum charts for monitoring process dispersion. PLOS ONE 10 (4):1–22.
  • Abujiya, M. R., M. H. Lee, and M. Riaz. 2016. Combined application of Shewhart and cumulative sum R chart for monitoring process dispersion. Quality and Reliability Engineering International, 32 (1):51–67.
  • Bisognin, C., and R. C. Lopes. 2009. Properties of seasonal long memory processes. Mathematical and Computer Modeling 49:1837–51.
  • Brook, D., and D. A. Evans. 1972. An approach to the probability distribution of the CUSUM Run Length. Biometrika 59 (3):539–49.
  • Busaba, J., S. Sukparungsee, and Y. Areepong. 2013. An analytical of average run length for first order of autoregressive observations on CUSUM procedure. International Journal of Applied Mathematics and Statistics 34:20–9.
  • Carlin, J. B., and A. P. Dempster. 1989. Sensitivity analysis of seasonal adjustments: empirical cases studies. Journal of the American Statistical Association 84:6–20.
  • Casas, I., and J. Gao. 2008. Econometric estimation in long-range dependent volatility models: theory and practice. Journal of Econometrics 147:72–83.
  • Champ, C. W., and S. E. Rigdon. 1991. A comparison of the Markov Chain and the Integral Equation approaches for evaluating the run length distribution of quality control charts. Communication in Statistics – Simulation 20:191–203.
  • Comte, F., and E. Renault. 1998. Long memory in continuous-time stochastic volatility models. Mathematical Finance 8:291–323.
  • Cisse, P. O., A. K. Diongue, and D. Guegan. 2016. Statistical properties of the seasonal fractionally integrated separable spatial autoregressive model. Afrika Statistika 11 (1):901–22.
  • Dugundji, J., and A. Granas. 2003. Fixed point theory. New York: Springer.
  • Duncan, A. J. 1974. Quality control and industrial statistics. 4th ed. Hornewood, Illinois: Richard D. Irwin, Inc.
  • Franses, P. H., and M. Ooms. 1997. A periodic long memory model for quarterly UK inflation. International Journal of Forecasting 13:117–26.
  • Fu, M. C., and J. Q. Hu. 1999. Efficient design and sensitivity analysis of control charts using Monte Carlo simulation. Management Science 45 (3):395–413.
  • Gan, F. F. 1992. Exact run length distribution for one-sided exponential CUSUM schemes. Statistica Sinica 2:297–312.
  • Goel, A. L., and S. M. Wu. 1973. Economically optimum design of CUSUM charts. Management Science 19:1271–82.
  • Granger, C. W. J., and R. Joyeux. 1980. An introduction to long-memory time series models and fractional differencing. Journal of Time Series Analysis 1 (1):15–29.
  • Granger, C. W. J. 1981. Some properties of time series data and their use in econometric model specification. Journal of Econometrics 16:121–30.
  • Hassler, U., and J. Wolters. 1995. Long memory in inflation rates. International evidence. Journal of Business and Economic Statistics 13:37–45.
  • Hawkins, D. M. 1981. A CUSUM for a scale parameter. Journal of Quality Technology 13 (4):228–31.
  • Hawkins, D. M., and D. H. Olwell. 1998. Cumulative Sum charts and charting for quality improvement. New York: Springer-Verlag.
  • Hosking, J. R. M. 1981. Fractional differencing. Biometrika. 68 (1):165–76.
  • Jacob, P. A., and P. A. W. Lewis. 1977. A mixed autoregressive-moving average exponential sequence and point process (EARMA 1,1). Advances in Applied Probability. 9 (1):87–104.
  • Lucas, J. M. 1976. The design and use of V-Mask control schemes. Journal of Quality Technology 8:1–12.
  • Lucas, J. M., and R. B. Crosier. 1982a. Fast initial response for CUSUM quality control schemes: Give your CUSUM A Head Start. Technometrics 24 (3):199–205.
  • Lucas, J. M., and M. S. Saccucci. 1990. Exponentially Weighted Moving Average control schemes properties and enhancements. Technometrics 32 (1):1–12.
  • Mititelu, G., Y. Areepong, S. Sukparungsee, and A. A. Novikov. 2010. Explicit analytical solutions for the average run length of CUSUM and EWMA chart. Contribution in Mathematics and Applications II East-West. Journal of Mathematics special 253–65.
  • Mohamed, I., and F. Hocine. 2003. Bayesian estimation of an AR(1) process with exponential white noise. Statistics 37:365–72.
  • Montanari, A., R. Rosso, and M. S. Taqqu. 2000. A seasonal fractional ARIMA model applied to the Nile river monthly at Aswan. Water Resources Research 36:1249–59.
  • Ooms, M. 1995. Flexible seasonal long memory and economic time series. Technical Report EI-95 15/A. Rotterdam: Econometric Institute, Erasmus University.
  • Page, E. S. 1954. Continuous inspection schemes. Biometrika. 42:243–254.
  • Pan, J. N., and S. T. Chen. 2008. Monitoring long-memory air quality data using ARFIMA model. Environmetrics 19:209–19.
  • Peerajit, W., Y. Areepong, and S. Sukparungsee. 2016. Numerical integral equation method of average run length of cumulative sum control chart for long memory process with ARFIMA model. Proceedings of the International MultiConference of Engineers and Computer Scientists (IMECS) with the Newswood Limited Publications, 2:852–55.
  • Pereira, I. M. S., and M. A. Turkrman. 2004. Bayesian prediction in threshold autoregressive models with exponential white noise. Sociedad de Estadisticae Investigacion Operativa Test. 13:45–64.
  • Petcharat, K., S. Sukparungsri, and Y. Areepong. 2015. Exact solution of the average run length for the cumulative sum chart for a moving average process of order q. Science Asia. 41:141–7.
  • Phanyaem, S., Y. Areepong, S. Sukparungsee, and G. Mititelu. 2014. Explicit formulas of average run length for ARMA(1, 1) process of CUSUM control chart. Far East Journal of Mathematical Sciences 90 (2):211–24.
  • Porter-Hudak, S. 1990. An application of the seasonal fractionally differenced model to the monetary aggregates. Journal of the American Statistical Association Vol. 85 (410):338–44.
  • Rabyk, L., and W. Schmid. 2016. EWMA control charts for detecting changes in the mean of a long-memory process. Metrika 79 (3):267–301.
  • Ramjee, R. 2000. Quality control charts and persistent processes. Ph.D. [dissertation]. Hoboken, New Jersey: Stevens Institute of Technology.
  • Ramjee, R., N. Crato, and B. K. Ray. 2002. A note on moving average forecasts of long memory processes with an application to quality control. International Journal of Forecasting 18:291–97.
  • Ray, B. K. 1993. Long-range forecasting of IBM product revenues using a seasonal fractionally differenced ARMA model. International Journal of Forecasting 9:255–69.
  • Reynolds, M. R., Jr. 1975. Approximations to the average run length in cumulative sum control charts. Technometrics 17:65–71.
  • Robert, S. W. 1959. Control Chart Test Based on Geometric Moving Averages. Technometrics 1:239–50.
  • Sanusi, R. A., M. R. Abujiya, M. Riaz, and N. Abbas. 2017. Combined shewhart cusum charts using auxiliary variable. Computers and Industrial Engineering 105:329–37.
  • Shitan, M. 2008. Fractionnaly Intergrated Separable Spatial Autoregressive (FISSAR) model and some of its properties. Communications in Statistics-Theory and Methods 37:1266–73.
  • Tsai, H. 2009. On continuous-time autoregressive fractionally integrated moving average processes. Bernoulli Journal 15 (1):178–94.
  • Tsai, H., and K. S. Chan. 2005a. Maximum likelihood estimation of linear continuous time long memory processes with discrete time data. Journal of the Royal Statistical Society Series B 67:703–16.
  • Tsai, H., and K. S. Chan. 2005b. Quasi-maximum likelihood estimation for a class of continuous-time long-memory processes. Journal of Time Series Analysis 26:691–713.
  • VanBrackle, L., and M. R. Reynolds. 1997. EWMA and CUSUM control charts in the presence of correlation. Communications in Statistics-Simulation and Computation 26:979–1008.
  • Venkateshwara, B. R., L. D. Ralph, and J. P. Joseph. 2001. Uniqueness and convergence of solutions to average run length integral equations for cumulative sums and other control charts. IIE Transactions 33:463–469.
  • Woodall, W. H. 1985. The statistical design of quality control charts. The Statistician, 34 (2):155–160.
  • Yashchin, E. 1993. Performance of CUSUM control schemes for serially correlated observations. Technometrics 35:37–52.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.