The random intrinsic fast initial response of two-sided CUSUM charts

A two-sided cumulative sum (CUSUM) chart may be defined as a joint plot of the successive values of a pair of upper and lower one-sided cumulative sum statistics. The standard procedure used to initialize these statistics consists in assigning a fixed starting value to each one of them. Each starting value may be taken equal to either zero or a fixed constant chosen to provide a fast initial response to the chart. In this article, the performance of a two-sided CUSUM chart is analyzed assuming that the pair of upper and lower one-sided statistics is initialized at a random point which follows a suitable defined joint stable distribution for the chart. This policy may be called the random intrinsic fast initial response (RIFIR) starting policy for the two-sided CUSUM chart. The article provides an algorithm to compute the stable distribution corresponding to the RIFIR starting policy as well as methods to evaluate the run-length distributions for in-control and out-of control situations. An important consequence of the stability of the RIFIR starting distribution is that, if no level shifts and no false alarms have occurred for a sufficiently large time interval (0, tau), the CUSUM chart behaves after tau as if the RIFIR initialization was used at tau, no matter how the chart was initialized at 0.