Average Run Lengths for CUSUM Schemes When Observations Are Exponentially Distributed

Abstract

Page (1954) originally noted that it is possible to find an integral equation whose solution gives average run lengths for one-sided CUSUM schemes. Lucas and Crosier (1982), for the case of normally distributed observations, have obtained numerical solutions to Page's integral equation and used these in their study of so called fast-initial-response CUSUM charts. In this article we show that for the case of exponentially distributed observations, the Page equation can be solved without resorting to approximations. We then provide some tables of average run lengths for the exponential case and comment on an application of exponential CUSUM charts to controlling the intensity of a Poisson process.

KEY WORDS:

• Average run length
• CUSUM scheme
• Exponential distribution
• Fast initial response
• Quality control
• Poisson process