Explicit Formulas and Numerical Integral Equation of ARL for SARX(P,r)_L Model Based on CUSUM Chart

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