Asymptotically optimal pointwise and minimax change-point detection for general stochastic models with a composite post-change hypothesis

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A weighted Shiryaev–Roberts change detection procedure is shown to approximately minimize the expected delay to detection as well as higher moments of the detection delay among all change-point detection procedures with the given low maximal local probability of a false alarm within a window of a fixed length in pointwise and minimax settings for general non-i.i.d. data models and for the composite post-change hypothesis when the post-change parameter is unknown. We establish very general conditions for models under which the weighted Shiryaev–Roberts procedure is asymptotically optimal. These conditions are formulated in terms of the rate of convergence in the strong law of large numbers for the log-likelihood ratios between the “change” and “no-change” hypotheses, and we also provide sufficient conditions for a large class of ergodic Markov processes. Examples related to multivariate Markov models where these conditions hold are given.