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Asymptotically optimal pointwise and minimax quickest change-point detection for dependent data

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  • Serguei Pergamenchtchikov

    (LMRS, CNRS - University of Rouen
    Lab. SSP-QF, Tomsk State University)

  • Alexander G. Tartakovsky

    (AGT StatConsult)

Abstract

We consider the quickest change-point detection problem in pointwise and minimax settings for general dependent data models. Two new classes of sequential detection procedures associated with the maximal “local” probability of a false alarm within a period of some fixed length are introduced. For these classes of detection procedures, we consider two popular risks: the expected positive part of the delay to detection and the conditional delay to detection. Under very general conditions for the observations, we show that the popular Shiryaev–Roberts procedure is asymptotically optimal, as the local probability of false alarm goes to zero, with respect to both these risks pointwise (uniformly for every possible point of change) and in the minimax sense (with respect to maximal over point of change expected detection delays). The 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, specifically as a uniform complete convergence of the normalized log-likelihood ratio to a positive and finite number. We also develop tools and a set of sufficient conditions for verification of the uniform complete convergence for a large class of Markov processes. These tools are based on concentration inequalities for functions of Markov processes and the Meyn–Tweedie geometric ergodic theory. Finally, we check these sufficient conditions for a number of challenging examples (time series) frequently arising in applications, such as autoregression, autoregressive GARCH, etc.

Suggested Citation

  • Serguei Pergamenchtchikov & Alexander G. Tartakovsky, 2018. "Asymptotically optimal pointwise and minimax quickest change-point detection for dependent data," Statistical Inference for Stochastic Processes, Springer, vol. 21(1), pages 217-259, April.
    • Handle: RePEc:spr:sistpr:v:21:y:2018:i:1:d:10.1007_s11203-016-9149-x
    DOI: 10.1007/s11203-016-9149-x
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    References listed on IDEAS

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    1. Galtchouk, L. & Pergamenshchikov, S., 2013. "Uniform concentration inequality for ergodic diffusion processes observed at discrete times," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 91-109.
    2. Paul D. Feigin & Richard L. Tweedie, 1985. "Random Coefficient Autoregressive Processes:A Markov Chain Analysis Of Stationarity And Finiteness Of Moments," Journal of Time Series Analysis, Wiley Blackwell, vol. 6(1), pages 1-14, January.
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    Cited by:

    1. Alexander G. Tartakovsky, 2023. "Quick and Complete Convergence in the Law of Large Numbers with Applications to Statistics," Mathematics, MDPI, vol. 11(12), pages 1-30, June.
    2. Pergamenchtchikov, Serguei M. & Tartakovsky, Alexander G. & Spivak, Valentin S., 2022. "Minimax and pointwise sequential changepoint detection and identification for general stochastic models," Journal of Multivariate Analysis, Elsevier, vol. 190(C).
    3. Pergamenchtchikov, Serguei & Tartakovsky, Alexander G., 2019. "Asymptotically optimal pointwise and minimax change-point detection for general stochastic models with a composite post-change hypothesis," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    4. Savas Dayanik & Kazutoshi Yamazaki, 2022. "Detection and identification of changes of hidden Markov chains: asymptotic theory," Statistical Inference for Stochastic Processes, Springer, vol. 25(2), pages 261-301, July.

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