An Approximation to the Average Run Length of CUSUM Control Chart for Long Memory under Fractionally Integrated Process with Exogenous Variable
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 is 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 by using the Numerical Integral Equation (NIE) method for long-memory process in case of exponential white noise. The NIE method approximate solutions are derived using the Gauss-Legendre quadrature rule technique. For the long memory process, the process is derived from the fractionally integrated with the exogenous variable model, which details the process depends on fractional differencing. This approximation ARL by 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.