Exact solution of the average run length for the cumulative sum chart for a moving average process of order q

Abstract

In this paper we use a Fredholm integral equation approach to derive an explicit formula for the average run length (ARL) of a cumulative sum (CUSUM) chart for random observations described by a moving average process of order q (MA(q)) with exponential white noise. We compare the computational times required for calculating the ARL from our exact formula with the computational times required for solving the Fredholm integral equations using a Gauss-Legendre numerical scheme. We find that the computational times are approximately 1 s for the explicit formula and approximately 13 min for the numerical integration scheme.