Performance analysis of sequential tests between Poisson processes

The problem of performance computation for sequential tests between Poisson processes is considered. The average sample numbers and error probabilities of the sequential probability ratio test (SPRT) between two homogeneous Poisson processes are known to solve certain delay-differential equations (DDEs). Exact, numerically stable solutions to these DDEs are developed here, and their asymptotic properties are explored. These solutions are seen to be superior to earlier solutions of Dvoretsky, Kiefer, and Wolfowitz (1953), which suffer from severe numerical instability in some ranges of parameters of interest in applications. The application of these results is illustrated in the problem of performance approximation for the cumulative sum (CUSUM) quickest detection procedure.