Computation of ARL on a CUSUM Control Chart for a Long-Memory Seasonal Autoregressive Fractionally Integrated Process with Exogenous Variables

Abstract

The aim of this study is to derive explicit formulas for the average run length (ARL) of a long-memory seasonal autoregressive fractionally integrated model with exogenous variables process on a cumulative sum (CUSUM) control chart with exponential white noise. The efficiency of the proposed explicit formulas was compared with a numerical integral equation (NIE) method via the relative percentage change to determine the accuracies of the ARL derivations. Although both methods were in good agreement, only 1 second of CPU time was required for the explicit formulas while 32-35 minutes were required for the NIE method. Some application examples using real datasets are provided to the detection ability of both methods.