Average Run Length with a Practical Investigation of Estimating Parameters of the EWMA Control Chart on the Long Memory AFRIMA Process
Keywords:
ARFIMA process, average run length, EWMA and CUSUM control chart, integral equationAbstract
An appropriate control chart for practical observations should be designed from optimal parameters. In this research, the main objectives are to estimate the optimal smoothing parameter of the EWMA control chart and fractional differencing parameter to evaluate the Average Run Length (ARL) and compare among analytical EWMA ARL, numerical EWMA ARL, and analytical CUSUM ARL. Also, the analytical EWMA ARL is derived and numerical EWMA ARL is evaluated and illustrated. The time intervals in days between explosions in mines in Great Britain during 1875 to 1951 are an example of practical observations of a long memory ARFIMA process with exponential white noise. The findings showed that the method for evaluating analytical EWMA ARL is an alternative for measurement of the efficiency of the EWMA control chart due to the good performance.
References
Areepong Y, Sukparungsee S. An integral equation approach to EWMA chart for detecting a change in lognormal distribution. Thail Stat. 2010; 8(1): 47-61.
Barbara O, Sílvia L, Valdério R. Invariance of the first difference in ARFIMA models. Comput Stat. 2006; 21: 445-461.
Barkoulas J, Baum C. Long memory and forecasting in Euroyen deposit rates. Finan Eng Japanese Markets. 1997; 4: 189-201.
Champ C, Rigdon S. A comparison of the Markov chain and the integral equation approaches for evaluating the run length distribution of quality control charts. Commun Stat-Simul. 1991; 20: 191-203.
Charles C, Jeh-Nan P. Evaluating environmental performance using statistical process control techniques. Eur J Oper Res. 2002; 139: 68-83.
Chris F, Washington U. R port by Fritz Leisch at TU Wien; since 2003-12: Martin Maechler; fdGPH, fdSperio, etc by Valderio Reisen and Artur Lemonte. (2012). fracdiff: Fractionally differenced ARIMA aka ARFIMA(p,d,q) models. R package version 1.4-2. https://CRAN.R-project.org/package=fracdiff.
Diane E, John D, Lawrence L. The distribution of the Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling test statistics for exponential populations with estimated parameters. Commun Stat Simul Comput. 2008; 37: 1396-1421.
Duan W, Chen Y. Goodness-of-fit test-statistics on Gaussian and exponential reliability data. IEEE Transactions on Reliability, 1983; 32(5): 492-495.
Fawaz B, Azhar H, Mohammad O. Evaluation of dissolved heavy metals in water of the Sungai Semenyih (Peninsula Malaysia) using environmetric methods. Sains Malays. 2016; 45(6): 841-852.
Geweke J, Porter-Hudak S. The estimation and application of long-memory time series models. J Time Ser Anal. 1983; 4: 221-238.
Granger J, Joyeux R. An introduction to long memory time series models and fractional differencing. J Time Ser Anal. 1980; 1(1): 15-39.
Haslett J, Raftery E. Space-time modelling with long-memory dependence: assessing ireland’s wind power resource (with Discussion). Appl Stat. 1989; 38: 1–50.
Hocine F. Testing on the first-order autoregressive model with contaminated exponential white noise finite sample case. Discussions Math Prob Stat. 2001; 21: 11-20.
Hosking M. Fractional differencing. Biometrika. 1981; 68(1): 165-176.
Jeh-Nan Pan, Su-Tsu Chen. Monitoring long-memory air quality data using ARFIMA model, Environmetrics. 2008; 19: 209-219.
Kharab A, Guenther B. An Introduction to Numerical Methods: A MATLAB Approach, 3rd edition, USA: CRC Press; 2012.
Kurita T. A forecasting model for Japan’s unemployment rate. Eur J Bus Econ. 2010; 3(5): 127-134.
Lee Y, Khoo C, Yap Y. A Comparison between the standard deviation of the run length (SDRL) performance of optimal EWMA and optimal CUSUM charts. J Qual Meas Anal. 2013; 9(1): 1-8.
Lim S, Lim C, Pauline W. 2008. ARIMA and integrated ARFIMA model for forecasting air pollution index in Shah Alam, Selangor. Malay J Anal Sci. 2008; 12(1): 257-263.
Liubov R, Wolfgang S. EWMA control charts for detecting changes in the mean of a long-memory process. Metrika. 2016; 79:267-301.
Lobato I, Velasco C. A simple and general test for white noise. Econometric Society, Latin-America Meetings. 2004; 1(112): 1-15.
Maguire A, Pearson S, Wynn A. The time intervals between industrial accidents. Biometrika 1952; 39 (1-2): 168-180.
Matheus G, Dmitry P. Numerical Mathematics, Sudbury (Massachusetts): Jones and Bartlett, Boston; 2008.
Mititelu G, Areepong Y, Sukparungsee S, Novikov A. Explicit analytical solutions for the average run length of CUSUM and EWMA charts. East-West J Math. 2010; 1: 253-265.
Muhammad A. Exponentially weighted moving average control charts for monitoring ambient ozone levels in Muscat. Amer J Theor Appl Stat. 2015; 4(4): 254-257.
Padhan C. Application of ARIMA model for forecasting agricultural productivity in India. J Agric Soc Sci. 2012; 8: 50-56.
Palma W, Chan N. Estimation and forecasting of long-memory processes with missing values. J Forecast. 1997; 16: 395-410.
Petar C and Sanja C. Optimization methods of EWMA statistics. Acta polytech. Hung. 2011; 8(5): 73-87.
Peter W. normwhn.test: normality and white noise testing. R package version 1.0. https://CRAN.R-project.org/package=normwhn.test. (2012).
Polunchenko S, Sokolov G, Tartakovsky G. Optimal design and analysis of the exponentially weighted moving average chart for exponential data. Sri Lankan J Appl Stat. 2014; 5(4): 57-80.
R Core Team. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org. (2016).
Samir S, Issam D. Comparative study on forecasting accuracy among moving average models with simulation and PALTEL stock market data in palestine. Int J Theor Appl Stat. 2013; 2(6): 202-209.
Saravanan A, Nagarajan P. Implementation of quality in bottle manufacturing industry. Int J Eng Sci Tech. 2013; 5(12): 335-340.
Suppanunta R. Forecasting model of RSS3 price in futures market. Kasetsart Univ J Econ. 2009; 16(1): 54-74.
Suriyakat W, Areepong Y, Sukparungsee S, Mititelu G. On EWMA procedure for an AR(1) observations with exponential white noise. Int J Pure Appl Math. 2012; 77: 73-83.
Barbara O, Sílvia L, Valdério R. Invariance of the first difference in ARFIMA models. Comput Stat. 2006; 21: 445-461.
Barkoulas J, Baum C. Long memory and forecasting in Euroyen deposit rates. Finan Eng Japanese Markets. 1997; 4: 189-201.
Champ C, Rigdon S. A comparison of the Markov chain and the integral equation approaches for evaluating the run length distribution of quality control charts. Commun Stat-Simul. 1991; 20: 191-203.
Charles C, Jeh-Nan P. Evaluating environmental performance using statistical process control techniques. Eur J Oper Res. 2002; 139: 68-83.
Chris F, Washington U. R port by Fritz Leisch at TU Wien; since 2003-12: Martin Maechler; fdGPH, fdSperio, etc by Valderio Reisen and Artur Lemonte. (2012). fracdiff: Fractionally differenced ARIMA aka ARFIMA(p,d,q) models. R package version 1.4-2. https://CRAN.R-project.org/package=fracdiff.
Diane E, John D, Lawrence L. The distribution of the Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling test statistics for exponential populations with estimated parameters. Commun Stat Simul Comput. 2008; 37: 1396-1421.
Duan W, Chen Y. Goodness-of-fit test-statistics on Gaussian and exponential reliability data. IEEE Transactions on Reliability, 1983; 32(5): 492-495.
Fawaz B, Azhar H, Mohammad O. Evaluation of dissolved heavy metals in water of the Sungai Semenyih (Peninsula Malaysia) using environmetric methods. Sains Malays. 2016; 45(6): 841-852.
Geweke J, Porter-Hudak S. The estimation and application of long-memory time series models. J Time Ser Anal. 1983; 4: 221-238.
Granger J, Joyeux R. An introduction to long memory time series models and fractional differencing. J Time Ser Anal. 1980; 1(1): 15-39.
Haslett J, Raftery E. Space-time modelling with long-memory dependence: assessing ireland’s wind power resource (with Discussion). Appl Stat. 1989; 38: 1–50.
Hocine F. Testing on the first-order autoregressive model with contaminated exponential white noise finite sample case. Discussions Math Prob Stat. 2001; 21: 11-20.
Hosking M. Fractional differencing. Biometrika. 1981; 68(1): 165-176.
Jeh-Nan Pan, Su-Tsu Chen. Monitoring long-memory air quality data using ARFIMA model, Environmetrics. 2008; 19: 209-219.
Kharab A, Guenther B. An Introduction to Numerical Methods: A MATLAB Approach, 3rd edition, USA: CRC Press; 2012.
Kurita T. A forecasting model for Japan’s unemployment rate. Eur J Bus Econ. 2010; 3(5): 127-134.
Lee Y, Khoo C, Yap Y. A Comparison between the standard deviation of the run length (SDRL) performance of optimal EWMA and optimal CUSUM charts. J Qual Meas Anal. 2013; 9(1): 1-8.
Lim S, Lim C, Pauline W. 2008. ARIMA and integrated ARFIMA model for forecasting air pollution index in Shah Alam, Selangor. Malay J Anal Sci. 2008; 12(1): 257-263.
Liubov R, Wolfgang S. EWMA control charts for detecting changes in the mean of a long-memory process. Metrika. 2016; 79:267-301.
Lobato I, Velasco C. A simple and general test for white noise. Econometric Society, Latin-America Meetings. 2004; 1(112): 1-15.
Maguire A, Pearson S, Wynn A. The time intervals between industrial accidents. Biometrika 1952; 39 (1-2): 168-180.
Matheus G, Dmitry P. Numerical Mathematics, Sudbury (Massachusetts): Jones and Bartlett, Boston; 2008.
Mititelu G, Areepong Y, Sukparungsee S, Novikov A. Explicit analytical solutions for the average run length of CUSUM and EWMA charts. East-West J Math. 2010; 1: 253-265.
Muhammad A. Exponentially weighted moving average control charts for monitoring ambient ozone levels in Muscat. Amer J Theor Appl Stat. 2015; 4(4): 254-257.
Padhan C. Application of ARIMA model for forecasting agricultural productivity in India. J Agric Soc Sci. 2012; 8: 50-56.
Palma W, Chan N. Estimation and forecasting of long-memory processes with missing values. J Forecast. 1997; 16: 395-410.
Petar C and Sanja C. Optimization methods of EWMA statistics. Acta polytech. Hung. 2011; 8(5): 73-87.
Peter W. normwhn.test: normality and white noise testing. R package version 1.0. https://CRAN.R-project.org/package=normwhn.test. (2012).
Polunchenko S, Sokolov G, Tartakovsky G. Optimal design and analysis of the exponentially weighted moving average chart for exponential data. Sri Lankan J Appl Stat. 2014; 5(4): 57-80.
R Core Team. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org. (2016).
Samir S, Issam D. Comparative study on forecasting accuracy among moving average models with simulation and PALTEL stock market data in palestine. Int J Theor Appl Stat. 2013; 2(6): 202-209.
Saravanan A, Nagarajan P. Implementation of quality in bottle manufacturing industry. Int J Eng Sci Tech. 2013; 5(12): 335-340.
Suppanunta R. Forecasting model of RSS3 price in futures market. Kasetsart Univ J Econ. 2009; 16(1): 54-74.
Suriyakat W, Areepong Y, Sukparungsee S, Mititelu G. On EWMA procedure for an AR(1) observations with exponential white noise. Int J Pure Appl Math. 2012; 77: 73-83.
Downloads
Published
2018-07-19
How to Cite
Sunthornwat, R., Areepong, Y., & Sukparungsee, S. (2018). Average Run Length with a Practical Investigation of Estimating Parameters of the EWMA Control Chart on the Long Memory AFRIMA Process. Thailand Statistician, 16(2), 190–202. Retrieved from https://ph02.tci-thaijo.org/index.php/thaistat/article/view/135562
Issue
Section
Articles
