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 Cumulative Sum (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 Sequential Probability Ratio Test (SPRT). The characteristics considered include the entire Run Length distribution and all of the corresponding moments, starting from the zero-state average run length. A particular practical benefit of our result is that it enables the in-control and out-of-control characteristics of the CUSUM Run Length to be computed concurrently. Moreover, owing to the equivalence of the CUSUM chart to a sequence of SPRTs, the Average Sample Number and Operating Characteristic 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 that one may choose to devise to carry out the actual computations. Appl. Stochastic Models Bus. Ind. 2016A. S. POLUNCHENKO and the SPRT; the possibility of such an extension was previously entertained in [23, Section 5]. Specifically, in this work, we employ Wald's [1] LR identity and establish a connection between a host of in-control characteristics of the CUSUM Run Length and their out-of-control counterparts. The connection is in the form of coupled integral (renewal) equations, and the derivation utilizes the well-known observation first made by Page [7] that the CUSUM chart is equivalent to repetitive application of the SPRT (with properly selected initial score and control bounds). The Run Length characteristics considered include the entire distribution and all of the corresponding moments, starting from the standard zero-state Average Run Length (ARL). On the practical side, the obtained connection enables concurrent evaluation of the in-control and out-of-control characteristics of the CUSUM Run Length. This would double up the efficiency of any numerical method that one may devise to compute the performance of the CUSUM chart (through solving the corresponding integral equations). The efficiency improvement would be of an even greater magnitude for the two-sided CUSUM chart, also proposed by Page [7, Section 3]. Moreover, since the CUSUM chart is equivalent to a sequence of SPRTs, the Average Sample Number (ASN) and the Operating Characteristic (OC) functions of an SPRT under the null and under the alternative can all be computed simultaneously as well, again with the aid of the main result obtained in the sequel. Hence, in a sense, this work is an attempt to bridge the gap between the theory and applications of sequential analysis.It is worth recalling that the need to evaluate the performance of the CUSUM chart (or that of the SPRT, or any other control chart for that matter) numerically is dictated by the fact that the corresponding characteristics (e.g., the zerostate A...

Some theoretical comments regarding the run-length properties of the synthetic and runs-rules X¯ monitoring schemes – part 1: zero-state

“…There is some indication that current publications are considering both modes; see Haq and Khoo (2016) for another example where both states were reported for their newly proposed auxiliary-information-based synthetic chart. However, unfortunately, other recent publications only focus on one mode without even making any mention of the other mode; see for example Polunchenko (2016) where only the zero-state is mentioned.…”