An approximation to the average run length of cumulative sum control chart for long memory under fractionally integrated process with exogenous variable
คำสำคัญ:Numerical Integral Equation (NIE) method, Exponential white noise, Long memory, CUSUM control chart, Average Run Length (ARL)
Cumulative sum (CUSUM) control chart is widely used in industries for the detection of small and moderate shifts in the process. Evaluation of the Average Run Length (ARL) plays an important role in the performance comparison of the control chart. Approximated ARL with the Numerical Integral Equation (NIE) method calculated by solving the system of linear equations and concept of integration based on the partition and summation the area under the curve of a function. The main purpose of this paper is to approximate the ARL of the CUSUM control chart using the Numerical Integral Equation (NIE) method for long-memory process in case of exponential white noise. The NIE method approximate solutions are derived by the Gauss-Legendre quadrature rule technique. For the long memory, the process is derived from the fractionally integrated with the exogenous variable model, which details the process depends on fractional differencing. This ARL approximation using NIE method is shown to be in good alternative compared with the explicit formula. An obvious extension is to other control charts for long memory under the fractionally integrated with the exogenous variable process, and hopefully, this work will encourage real-world applications such as finance economics and agriculture.
Page ES. Continuous inspection schemes. Biometrika 1954;41:100-14.
Gan FF. An optimal design of CUSUM quality control charts. J Qual Technol 1991;23(4):279-86.
Luceno A, Puig-Pey J. The random intrinsic fast initial response of one-sided CUSUM charts. J App Stat 2006;33(2):189-201.
Wu Z, Wang QN. A single CUSUM chart using a single observation to monitor a variable. Int J Prod Res 2007;45(3):719-41.
Ryu JH, Wan H, Kim S. Optimal design of a CUSUM chart for a mean shift of unknown size. J Qual Technol 2010;42(3):311-26.
Johnson RA, Bagshaw M. The effect of serial correlation on the performance of CUSUM tests. Technometrics 1974; 16(1):103-12.
Lu CW, Reynolds MRJ. Cusum charts for monitoring an autocorrelated process. J Qual Technol 2001; 33:316-34.
Kim SH, Alexopoulos C, Tsui KL, Wilson JR. A distribution-free tabular CUSUM chart for autocorrelated data. IIE Trans 2007;39:317-30.
Jacob PA, Lewis PAW. A mixed autoregressive-moving average exponential sequence and point process (EARMA 1,1). Adv Appl Probab 1977;9(1):87-104.
Mohamed I, Hocine F. Bayesian estimation of an AR(1) process with exponential white noise. AJTAS 2003; 37(5):365-372.
Pereira IMS, Turkrman MA. Bayesian prediction in threshold autoregressive models with exponential white noise. SEIO 2004;13:45-64.
Granger CWJ, Joyeux R. An introduction to long memory time series models and fractional differencing. J Time Ser Anal 1980;1(1):15-29.
Hosking JRM. Fractional differencing. Biometrika 1981;68(1):165-76.
Marmol F, Velasco C. Trend stationary versus long-range dependence in time series analysis. J Econom 2002;108:25-42.
Vanbrackle L, Reynolds MR. EWMA and CUSUM control charts in the presence of correlation. Commun Stat-Simul C 1997;26:979-1008.
Sukparungsee S, Novikov AA. On EWMA procedure for detection of a change in observations via martingale approach. Int J Appl Sci 2006;6:373-80.
Busaba J, Sukparungsee S, Areepong Y. Numerical approximations of average run length for AR(1) on exponential CUSUM. In: proceedings of the International MutiConference of Engineers and Computer Scientists, March 14-16; 2012; Hongkong; 2012. p. 1268-1273.
Phanyaem S, Areepong Y, Sukparungsee S, Mititelu G. Explicit formulas of average run length for ARMA(1,1). IJAMAS 2013;43:392-405.
Phanyaem S, Areepong Y, Sukparungsee S. Numerical integration of average run length of CUSUM control chart for ARMA process. IJAPM 2014;4(4):232-5.
Paichit P, Areepong Y, Sukparungsee S, Average run length of control chart for ARX(1) process with exponential white noise. IJAPM 2016;12(3):2143-53.
Peerajit W, Areepong Y, Sukparungsee S. Numerical integral equation method for ARL of CUSUM chart for long-memory process with non-seasonal and seasonal ARFIMA models. Thai Stat 2018;6(1):26-37.
Baillie RT. Long memory processes and fractional integration in econometrics. J Econom 1996;73(1):5-59.
Beran J, Feng Y, Ghosh S, Kulik R. Long memory processes: probabilistic properties and statistical methods. London: Springer; 2013.
Page ES. Controlling the standard deviation and warning lines by CUSUM. Technometrics 1963;5(3):307-15.
Ramjee R, Crato N, Ray BK. A note on moving average forecasts of long memory processes with an application to quality control. Int J Forecast 2002;18:291-7.
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