Analytical Assessment of the DMEWMA Control Chart for Detecting Shifts in ARI and ARFI Models with Applications

This paper presents an analytical study of the double modified exponentially weighted moving average (DMEWMA) control chart for monitoring autoregressive integrated (ARI) and autoregressive fractionally integrated (ARFI) processes, with emphasis on its symmetry properties. The explicit formulas for the average run length (ARL) are derived using Fredholm integral equations of the second kind, with existence and uniqueness established via Banach’s fixed-point theorem. The numerical approximations are obtained through the numerical integral equation (NIE) method, and simulations confirm that explicit ARL formulas and the NIE approach yield identical results, validating the theoretical derivations. The results show that symmetry plays a dual role: it ensures performance symmetry in detection across short- and long-memory processes. Comparative studies indicate that for ARI processes, DMEWMA outperforms EWMA and MEWMA for small and moderate shifts, while for ARFI processes, it remains superior to EWMA but shows parity or slight inferiority to MEWMA under certain long-memory conditions. The applications to environmental and economic data illustrate the value of symmetrical control structures in providing robust, unbiased monitoring. With ARL computations completed in under 0.001 s, the DMEWMA chart demonstrates efficiency, balance, and versatility.