The Effectiveness of CUSUM Control Chart for Trend Stationary Seasonal Autocorrelated Data

Abstract

In this paper, the average run length (ARL) for the cumulative sum (CUSUM) control chart based on SAR(P)_L with trend process is developed using Fredholm integral equation approach. The numerical integral equation is used to approximate the ARL. Banach’s Fixed Point theorem is used to guarantee the existence and uniqueness of the solution. Finally, the accuracy of the proposed formulas is established by comparing them to the numerical integration method. The results from explicit formula and numerical integration show that the absolute percentage relative error is less than 0.3% In terms of computational time, the explicit formula can perform better than the numerical integration. In addition, real-world application of the derived explicit formulas was illustrated using the silver price data (USD/oz) to evaluate the ARL of SAR(P)_L process on a CUSUM control chart.