# Average Run Length with a Practical Investigation of Estimating Parameters of the EWMA Control Chart on the Long Memory AFRIMA Process

### Abstract

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.

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