Asymptotically Optimal Methods of Early Change-Point Detection

In this article the problem of early change-point detection is considered. The main difference from traditional sequential change-point detection problems consists in the nature of changes: we consider both gradual and abrupt changes and wish to detect instants of the beginning of these changes as soon as possible on condition that false alarms are few. We prove the theoretical informational inequalities for the main performance characteristics of early change-point detection methods that help us to consider the asymptotically optimal methods. Our criterion of asymptotic optimality does not coincide with traditional criteria of Shiryaev (1963 Shiryaev , A. N. ( 1963 ). Optimal Methods in Sequential Change-Point Detection Problems , Probability Theory and Its Applications 8 : 26 – 51 .[Crossref] , [Google Scholar]), Lorden (1971 Lorden , G. ( 1971 ). Procedures for Reacting to a Change in Distribution , Annals of Mathematical Statistics 42 : 1897 – 1908 .[Crossref] , [Google Scholar]), and Pollak (1985 Pollak , M. ( 1985 ). Optimal Detection of a Change in Distribution , Annals of Statistics 13 : 206 – 227 .[Crossref], [Web of Science ®] , [Google Scholar]) but allows us to consider both univariate and multivariate nonstationary stochastic models. Besides the theoretical analysis of univariate and multivariate models with changes, we present results of the Monte Carlo study of the proposed methods.