Asymptotically optimal methods of change-point detection for composite hypotheses

In this paper the problem of change-point detection for the case of composite hypotheses is considered. We assume that the distribution functions of observations before and after an unknown change-point belong to some parametric family. The true value of the parameter of this family is unknown but belongs to two disjoint sets for observations before and after the change-point, respectively. A new criterion for the quality of change-point detection is introduced. Modifications of generalized CUSUM and GRSh (Girshick-Rubin-Shiryaev) methods are considered and their characteristics are analyzed. Comparing these characteristics with an a priori boundary for the quality of change-point detection we establish asymptotic optimality of these methods when the family of distributions before the change-point consists of one element.