Lobachevskii Journal of Mathematics

, Volume 39, Issue 3, pp 424–432 | Cite as

The ARIMA(p,d,q) on Upper Sided of CUSUM Procedure

  • Lili Zhang
  • Piyapatr Busababodhin


In this paper we derive explicit formula for the average run length (ARL) of Cumulative Sum (CUSUM) control chart of autoregressive integrated moving average ARIMA(p,d,q) process observations with exponential white noise. The explicit formula are derived and the numerical integrations algorithm is developed for comparing the accuracy. We derived the explicit formula for ARL by using the Integral equations (IE) which is based on Fredholm integral equation. Then we proof the existence and uniqueness of the solution by using the Banach’s fixed point theorem. For comparing the accuracy of the explicit formulas, the numerical integration (NI) is given by using the Gauss-Legendre quadrature rule. Finally, we compare numerical results obtained from the explicit formula for the ARL of ARIMA(1,1,1) processes with results obtained from NI. The results show that the ARL from explicit formula is close to the numerical integration with an absolute percentage difference less than 0.3% with m = 800 nodes. In addition, the computational time of the explicit formula are efficiently smaller compared with NI.


Average run length Cumulative Sum Autoregressive integrated moving average 


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  1. 1.
    N. F. Zhang, “A statistical control chart for stationary process data,” Technometrics 40, 24–38 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    C. W. Lu and M. R. Reynolds, “EWMA control charts for monitoring the mean of autocorrelated processes,” J. Qual. Tech. 31, 166–188 (1999).CrossRefGoogle Scholar
  3. 3.
    M. Rosolowski and W. Schmid, “EWMA charts formonitoring the mean and the autocovariance of stationary processes,” Stat. Pap. 47, 595–603 (2006).CrossRefzbMATHGoogle Scholar
  4. 4.
    M. B. Vermaat, F. H. Van der Meulen, and R. Does, “Asymptotic behavior of the variance of the EWMA statistic for autoregressive processes,” Stat. Probab. Lett. 78, 1673–1682 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    C. C. Torng, P. H. Lee, H. S. Liao, and N. Y. Liao, “An economic design of double sampling X charts for correlated data using genetic algorithms,” Expert Syst. Appl. 36, 12621–12626 (2009).CrossRefGoogle Scholar
  6. 6.
    D. P. Gaver and P. A. W. Lewis, “First-order autoregressive Gamma sequences and point processes,” Adv. Appl. Prob. 12, 727–745 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    C. B. Bell and E. P. Smith, “Inference for non-negative autoregressive schemes,” Commun. Stat. Theory 15, 2267–2293 (1986).CrossRefzbMATHGoogle Scholar
  8. 8.
    J. Andel, “On AR(1) processeswith exponential white noise,” Commun. Stat. Theory 17, 1481–1495 (1988).CrossRefzbMATHGoogle Scholar
  9. 9.
    R. Davis and W. P. McCormick, “Estimation for first-order autoregressive processeswith positive or bounded innovations,” Stoch. Proc. Appl. 31, 237–250 (1989).CrossRefzbMATHGoogle Scholar
  10. 10.
    J. Andel and M. Garrido, “Bayesian analysis of non-negative AR(2) processes,” Statistics 22, 579–588 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    W. P. McCormick and G. Mathew, “Estimation for nonnegative autoregressive processes with an unknown location parameter,” J. Time Ser. Anal. 14, 71–92 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    D. Brook and D. A. Evans, “An approach to the probability distribution of CUSUM run lengths,” Biometrika 59, 539–548 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    T. J. Harris and W. H. Ross, “Statistical process control procedures for correlated observations,” Can. J. Chem. Eng. 69, 48–57 (1991).CrossRefGoogle Scholar
  14. 14.
    C. M. Mastrangelo and C. M. Montgomery, “SPC with correlated observations for the chemical and process industries,” Qual. Reliab. Eng. Int. 11, 79–89 (1995).CrossRefGoogle Scholar
  15. 15.
    L. N. Van Brackle and M. R. Reynolds, “EWMAand CUSUM control charts in the presence of correlation,” Comm. Stat. Simulat. Comput. 26, 979–1008 (1997).CrossRefGoogle Scholar
  16. 16.
    G. Mititelu, Y. Areepong, S. Sukparungsee, and A. A. Novikov, “Explicit analytical solutions for the average run length of CUSUM and EWMA charts,” East-West J.Math. 1, 253–265 (2010).MathSciNetzbMATHGoogle Scholar
  17. 17.
    J. Busaba, S. Sukparungsee, Y. Areepong, and G. Mititelu, “Numerical approximations of average run length for AR(1) on exponential CUSUM,” in Proceedings of the International MutiConference of Engineers and Computer Scientists, Hong Kong, March 2012, pp. 14–16.Google Scholar
  18. 18.
    K. Petcharat, Y. Areepong, S. Sukparungsee, and G. Mititelu, “Exact solution of average run length of EWMA chart for MA(q) processes,” Far East J.Math. Sci. 78, 291–300 (2013).zbMATHGoogle Scholar
  19. 19.
    S. Phanyaem, Y. Areepong, S. Sukparungsee, and G. Mittitelu, “Explicit formulas of average run length for ARMA(1,1),” Int. J. Appl.Math. Stat. 43, 392–405 (2013).MathSciNetGoogle Scholar
  20. 20.
    P. Suvimol, “Average run length of cumulative sum control charts for SARMA(1, 1)L models,” Thailand Stat. 15, 184–195 (2017).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Department of StatisticsChonnam National UniversityGwangjuKorea
  2. 2.Department of MathematicsMahasarakham UniversityMahasarakhamThailand

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