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A weighted Shiryaev-Roberts change detection procedure is shown toapproximately minimize the expected delay to detection as well as highermoments of the detection delay among all change-point detection procedures withthe given low maximal local probability of a false alarm within a window of afixed length in pointwise and minimax settings for general non-i.i.d. datamodels and for the composite post-change hypothesis when the post-changeparameter is unknown. We establish very general conditions for the models underwhich the weighted Shiryaev-Roberts procedure is asymptotically optimal. Theseconditions are formulated in terms of the rate of convergence in the strong lawof large numbers for the log-likelihood ratios between the "change" and"no-change" hypotheses, and we also provide sufficient conditions for a largeclass of ergodic Markov processes. Examples, where these conditions hold, aregiven.

Asymptotically Optimal Pointwise and Minimax Change-point Detection for General Stochastic Models With a Composite Post-Change Hypothesis