Minimax Sequential Testing for Poisson Processes
Hongwei Mei
TL;DR
The paper addresses robust sequential hypothesis testing for a Poisson process with two possible intensities $λ_0$ and $λ_1$ when the prior is unknown. It recasts the problem as a minimax Bayesian optimal stopping problem, derives an equivalent characterization of least favorable distributions, and proves a sufficient condition for their existence along with a practical numerical method. A key finding is that the least favorable prior need not be neutral (e.g., not necessarily $1/2$ even under symmetric costs), and the minimax optimal stopping boundary can be identified via an auxiliary boundary problem involving $α^*,β^*$. Numerical examples illustrate the method and demonstrate asymmetry in the minimax solution, offering a robust framework for real-time Poisson-process testing and potential extensions to more general jump processes.
Abstract
Suppose we observe a Poisson process in real time for which the intensity may take on two possible values $λ_0$ and $λ_1$. Suppose further that the priori probability of the true intensity is not given. We solve a minimax version of Bayesian problem of sequential testing of two simple hypotheses to minimize a linear combination of the probability of wrong detection and the expected waiting time in the worst scenario of all possible priori distributions. An equivalent characterization for the least favorable distributions is derived and a sufficient condition for the existence is concluded.