Analytical and numerical solutions of average run length integral equations for an EWMA control chart over a long memory SARFIMA process

The efficiency of a process, especially a long memory seasonal autoregressive fractional integral moving average (SARFIMA) process, has commonly been measured through the quality control chart. In this paper, a generalized long memory SARFIMA process of the exponentially weighted moving average (EWMA) control chart is carried out and shown. Also, analytical and numerical average run length (ARL) were designed to measure the efficiency of the EWMA control. Existence and uniqueness by the fixed point theory are proven for the analytical ARL. Error and convergence of numerical integration equations are also given for the numerical ARL. The findings indicated that the analytical ARL was evaluated more quickly and accurately than the numerical ARL. As a result, the analytical ARL is an alternative for measuring the efficiency of an EWMA control chart over a long memory SARFIMA process