Asymptotical Optimality of Change Point Detection With Unknown Discrete Post-Change Distributions

Abstract

The change point detection theory seeks to capture the changes as soon as possible after they occur, with low false alarm. We investigate this problem under the condition that the post-change distribution defined on a finite sample space is not available. We introduce a sequential version of universal hypothesis test across the curved boundary, and prove that this sequential test asymptotically achieves smaller average sample size than any other sequential test. Based on this sequential test, we propose two types of change point detection procedures, one is Lorden procedure and another is Shiryaev-Roberts procedur. Both of them are asymptotically optimal with their corresponding criterion. The substantial statistic properties are presented by simulations. Our results shed some light on optimal analysis in nonparametric change point detection procedures.