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

TL;DR

This paper establishes a mathematical connection between the in-control and out-of-control characteristics of the CUSUM chart and SPRT, enabling more efficient simultaneous computation of their performance metrics.

Contribution

It introduces a method using integral equations and Wald's likelihood ratio identity to compute CUSUM and SPRT characteristics concurrently, improving computational efficiency.

Findings

Allows simultaneous calculation of in- and out-of-control metrics.

Utilizes integral equations and likelihood ratio identity.

Enhances efficiency of performance evaluation methods.

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.