Explicit Formulas and Numerical Integral Equation of ARL for SARX(P,r)<sub>L</sub> Model Based on CUSUM Chart

Abstract

The Cumulative Sum (CUSUM) chart is widely used and has many applications in different fields such as finance, medical, engineering, and other fields. In real applications, there are many situations in which the observations of random processes are serially correlated, such as a hospital admission in the medical field, a share price in the economic field, or a daily rainfall in the environmental field. The common characteristic of control charts that has been used to evaluate the performance of control charts is the Average Run Length (ARL). The primary goals of this paper are to derive the explicit formula and develop the numerical integral equation of the ARL for the CUSUM chart when observations are seasonal autoregressive models with exogenous variable, SARX(P,r)_L with exponential white noise. The Fredholm Integral Equation has been used for solving the explicit formula of ARL, and we used numerical methods including the Midpoint rule, the Trapezoidal rule, the Simpson's rule, and the Gaussian rule to approximate the numerical integral equation of ARL. The uniqueness of solutions is guaranteed by using Banach's Fixed Point Theorem. In addition, the proposed explicit formula was compared with their numerical methods in terms of the absolute percentage difference to verify the accuracy of the ARL results and the computational time (CPU). The results obtained indicate that the ARL from the explicit formula is close to the numerical integral equation with an absolute percentage difference of less than 1%. We found an excellent agreement between the explicit formulas and the numerical integral equation solutions. An important conclusion of this study was that the explicit formulas outperformed the numerical integral equation methods in terms of CPU time. Consequently, the proposed explicit formulas and the numerical integral equation have been the alternative methods for finding the ARL of the CUSUM control chart and would be of use in fields like biology, engineering, physics, medical, and social sciences, among others.