Developing Average Run Length for Monitoring Changes in the Mean on the Presence of Long Memory under Seasonal Fractionally Integrated MAX Model

The cumulative sum (CUSUM) control chart can sensitively detect small-to-moderate shifts in the process mean. average run length (ARL) is a popular technique used to determine performance of chart. Recently, several researchers investigated processes on CUSUM by evaluating ARL using either Monte Carlo simulation or Markov chain. As these methods only yield approximate results, we developed solutions for exact explicit formulas based an integral equation (IE) studying running long-memory with exponential white noise. observations are derived from seasonal fractionally integrated MAX model while focusing X. existence and uniqueness solution calculating via were proved Banach's fixed-point theorem. accuracy percentage against obtained numerical IE method was greater than 99%, which indicates excellent agreement between two methods. An important conclusion this study that proposed could changes mean situation. Finally, illustrative case provided show efficacy involving real data.