A Note on Efficient Performance Evaluation of the Cumulative Sum Chart and the Sequential Probability Ratio Test

Abstract: We establish a simple connection between certain in-control characteristics of the CUSUM Run Length and their out-of-control counterparts. The connection is in the form of paired integral (renewal) equations. The derivation exploits Wald's likelihood ratio identity and the well-known fact that the CUSUM chart is equivalent to repetitive application of Wald's SPRT. The characteristics considered include the entire Run Length distribution and all of the corresponding moments, starting from the zero-state ARL. A particular practical benefit of our result is that it enables the in- and out-of-control characteristics of the CUSUM Run Length to be computed concurrently. Moreover, due to the equivalence of the CUSUM chart to a sequence of SPRTs, the ASN and OC functions of an SPRT under the null and under the alternative can all be computed simultaneously as well. This would double up the efficiency of any numerical method one may choose to devise to carry out the actual computations.