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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