- Volume 30 Issue 1
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Average run length calculation of the EWMA control chart using the first passage time of the Markov process
Markov 과정의 최초통과시간을 이용한 지수가중 이동평균 관리도의 평균런길이의 계산
- Park, Changsoon (Department of Applied Statistics, Chung-Ang University)
- 박창순 (중앙대학교 응용통계학과)
- Received : 2016.10.12
- Accepted : 2016.11.03
- Published : 2017.02.28
Many stochastic processes satisfy the Markov property exactly or at least approximately. An interested property in the Markov process is the first passage time. Since the sequential analysis by Wald, the approximation of the first passage time has been studied extensively. The Statistical computing technique due to the development of high-speed computers made it possible to calculate the values of the properties close to the true ones. This article introduces an exponentially weighted moving average (EWMA) control chart as an example of the Markov process, and studied how to calculate the average run length with problematic issues that should be cautioned for correct calculation. The results derived for approximation of the first passage time in this research can be applied to any of the Markov processes. Especially the approximation of the continuous time Markov process to the discrete time Markov chain is useful for the studies of the properties of the stochastic process and makes computational approaches easy.
Supported by : 한국연구재단
- Champ, C. W. and Rigdon, S. E. (1991). A comparison of the Markov chain and the integral equation approaches for evaluating the run length distribution of quality control charts, Communications in Statistics-Simulation and Computation, 20, 191-204. https://doi.org/10.1080/03610919108812948
- Chan, Y., Han, B., and Pascual, F. (2015). Monitoring the Weibull shape parameter with type II censored data, Quality and Reliability Engineering International, 31, 741-760. https://doi.org/10.1002/qre.1631
- Chang, T. C. and Gan, F. F. (1994). Optimal designs of one-sided EWMA charts for monitoring a process variance, Journal of statistical Computation and Simulation, 49, 33-48. https://doi.org/10.1080/00949659408811559
- Crowder, S. V. (1987a). A simple method for studying run-length distributions of exponentially weighted moving average charts, Technometrics, 29, 401-407.
- Crowder, S. V. (1987b). Average run length of exponentially weighted moving average charts, Journal of Quality Technology, 19, 161-164. https://doi.org/10.1080/00224065.1987.11979055
- Crowder, S. V. and Hamilton, M. D. (1992). An EWMA for monitoring a process standard deviation, Journal of Quality Technology, 24, 12-21. https://doi.org/10.1080/00224065.1992.11979369
- Gan, F. F. (1993). Exponentially weighted moving average control charts with reflecting boundaries, Journal of statistical Computation and Simulation, 46, 45-67. https://doi.org/10.1080/00949659308811492
- Gan, F. F. (1995). Joint monitoring of process mean and variance using exponentially weighted moving average control charts, Technometrics, 37, 446-453. https://doi.org/10.1080/00401706.1995.10484377
- Gan, F. F. (1998). Designs of one- and two-sided exponential EWMA charts, Journal of Quality Technology, 30, 55-69. https://doi.org/10.1080/00224065.1998.11979819
- Gan, F. F. and Chang, T. C. (2000). Computing average run lengths of exponential EWMA charts, Journal of Quality Technology, 32, 183-187. https://doi.org/10.1080/00224065.2000.11979989
- Gianino, A. B., Champ, C. W., and Rigdon, S. E. (1990). Solving integral equations by the collocation method. In ASA Proceedings of the Statistical Computing Section (pp. 101-102), American Statistical Association, Washington.
- Hunter, J. S. (1986). The exponentially weighted moving average, Journal of Quality Technology, 18, 203-210. https://doi.org/10.1080/00224065.1986.11979014
- Knoth, S. (2003). EWMA schemes with non-homogeneous transition kernels, Sequential Analysis, 22, 241-255. https://doi.org/10.1081/SQA-120025169
- Knoth, S. (2004). Fast initial response features for EWMA control charts, Statistical Papers, 46, 47-64.
- Knoth, S. (2005). Accurate ARL computation for EWMA-S2 control charts, Statistics and Computing, 15, 341-352. https://doi.org/10.1007/s11222-005-3393-z
- Knoth, S. (2007). Accurate ARL calculation for EWMA control charts monitoring simultaneously normal mean and variance, Sequential Analysis, 26, 251-264. https://doi.org/10.1080/07474940701404823
- Lucas, J. M. and Saccucci, M. S. (1990). Exponentially weighted moving average control schemes: properties and enhancements, Technometrics, 32, 1-12. https://doi.org/10.1080/00401706.1990.10484583
- MacGregor, J. F. and Harris, T. J. (1993). The exponentially weighted moving variance, Journal of Quality Technology, 25, 106-118. https://doi.org/10.1080/00224065.1993.11979433
- Nichols, M. D. and Padgett, W. J. (2005). A bootstrap control chart for Weibull percentiles, Quality and Reliability Engineering International, 22, 141-151.
- Park, C. (2007). An algorithm for the properties of the integrated process control with bounded adjustments and EWMA monitoring, International Journal of Production Research, 45, 5571-5587. https://doi.org/10.1080/00207540701325397
- Park, C., Lee, J., and Kim, Y. (2004). Economic design of a variable sampling rate EWMA chart, IIE Transactions, 36, 387-399. https://doi.org/10.1080/07408170490426116
- Park, C. and Reynolds, M. R. (1999). Economic design of a variable sampling rates X chart, Journal of Quality Technology, 31, 363-443. https://doi.org/10.1080/00224065.1999.11979943
- Park, C. and Reynolds, M. R. (2008). Economic design of an integrated process control procedure with repeated adjustments and EWMA monitoring, Journal of the Korean Statistical Society, 37, 155-174. https://doi.org/10.1016/j.jkss.2007.10.005
- Park, C. S. and Won, T. Y. (1996). Selection of the economically optimal parameters in the EWMA control chart, Korean Journal of Applied Statistics, 9, 91-109.
- Pascual, F. (2010). EWMA charts for the Weibull shape parameter, Journal of Quality Technology, 42, 400-416. https://doi.org/10.1080/00224065.2010.11917836
- Ramalhoto, M. F. and Morais, M. (1999). Shewhart control charts for the scale parameter of a Weibull control variable with fixed and variable sampling intervals, Journal of Applied Statistics, 26, 129-160. https://doi.org/10.1080/02664769922700
- Waldmann, K. H. (1986). Bounds for the distribution of the run length of geometric moving average charts, Applied Statistics, 35, 151-158. https://doi.org/10.2307/2347265