A Novel Numerical Integration Equation Approach using the Gauss-Legendre Quadrature to Approximate the Average Run Length of Time-Series Model Running on a CUSUM Control Chart

Herein, we present the comprehensive validation of an approximation of the average run length (ARL) for monitoring small-to-moderate changes in the process means of long-memory fractionally integrated moving average with exogenous variables (LFIMAX) processes with different types of long memory on a cumulative sum (CUSUM) control chart based on the numerical integration equation (NIE) method. The ARL approximation using the NIE method was obtained by resolving a system of linear equations and performing integration through the partitioning and summing of the area under the curve of a function produced from the Gauss-Legendre quadrature rule. The performances of the proposed NIE and an established analytical method were compared for mean shifts of varying sizes for LFIMAX processes on a CUSUM control chart. The numerical results indicate that the proposed ARL method performed comparably with the analytical method regarding percentage accuracy Acc(%)). Moreover, a small percentage relative deviation (DEV%) is indicated, i.e., a change in magnitude of less than 0.25 can be detected rapidly in all situations. A numerical example using real-world scenario data is also provided to illustrate the practicability of the proposed method.