Average Run Length of Cumulative Sum Control Charts for SARMA(1,1)L Models

Abstract

The cumulative sum (CUSUM) control chart is widely used in a great variety of practical applications such as finance, medicine, engineering, psychology and in other areas. There are many situations in which the process is serially correlation such as in the manufacturing industry, for example, the dynamics of the process will induce correlations in observations that are closely spaced in time. The average run length (ARL) is a traditional measurement of the performance of control chart. In this paper we derive explicit formula for the ARL of CUSUM control chart when observations are seasonal autoregressive and moving average, SARMA(1,1)_L process with exponential white noise. We use Fredholm integral equation approach to derive an explicit formula forthe ARL anduse the Gauss-Legendre quadrature rule to approximate the numerical integration which both methods based on the Banach’s fixed point theorem which is used to guarantee the existence and uniqueness of the solution. Finally, we compare numerical results obtained from the explicit formula for the ARL of SARMA(1,1)_Lprocesses with results obtained from a numerical solution of an integral equation for the ARL. The results show that the ARL from explicit formula is close to the numerical integration with an absolute percentage difference less than 0.1%. In addition, the explicit formula can reduce in the computational timebetter than the numerical integration.