A Novel Explicit ARL Formula via Integral Equation on DEWMA for Detecting Mean Shifts in the Autoregressive Process with Exogenous Variables

This study develops explicit formulas for the Average Run Length (ARL) of the Double Exponentially Weighted Moving Average (DEWMA) control chart under ARX(p,r) processes with exponential white noise. The proposed approach provides an efficient analytical framework for monitoring autocorrelated and non-normally distributed processes. The existence and uniqueness of the ARL solution are established using Banach’s fixed-point theorem, while the accuracy of the derived formulas is validated against Numerical Integral Equation (NIE) solutions obtained via the Midpoint Rule, Gauss-Legendre Quadrature Rule, Trapezoidal Rule, and Simpson’s Rule. The numerical results show that the explicit formulas produce ARL values that are in close agreement with those obtained from the NIE methods, while substantially reducing computational burden. Furthermore, performance comparisons based on the Expected Average Run Length (EARL), Average Extra Quadratic Loss (AEQL), and Performance Comparison Index (PCI) reveal that the DEWMA control chart is more effective than the EWMA control chart in detecting small and moderate shifts. A real-data application to natural gas prices, incorporating West Texas Intermediate (WTI) crude oil prices as an exogenous variable, further demonstrates the practical usefulness and effectiveness of the proposed methodology for process monitoring in the presence of autocorrelation and external influences.