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Truncated Sequential Change‐point Detection based on Renewal Counting Processes

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  • ALLAN GUT
  • JOSEF STEINEBACH

Abstract

The typical approach in change‐point theory is to perform the statistical analysis based on a sample of fixed size. Alternatively, one observes some random phenomenon sequentially and takes action as soon as one observes some statistically significant deviation from the “normal” behaviour. Based on the, perhaps, more realistic situation that the process can only be partially observed, we consider the counting process related to the original process observed at equidistant time points, after which action is taken or not depending on the number of observations between those time points. In order for the procedure to stop also when everything is in order, we introduce a fixed time horizon n at which we stop declaring “no change” if the observed data did not suggest any action until then. We propose some stopping rules and consider their asymptotics under the null hypothesis as well as under alternatives. The main basis for the proofs are strong invariance principles for renewal processes and extreme value asymptotics for Gaussian processes.

Suggested Citation

  • Allan Gut & Josef Steinebach, 2002. "Truncated Sequential Change‐point Detection based on Renewal Counting Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 29(4), pages 693-719, December.
    • Handle: RePEc:bla:scjsta:v:29:y:2002:i:4:p:693-719
    DOI: 10.1111/1467-9469.00313
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    Cited by:

    1. Michael Messer & Gaby Schneider, 2017. "The shark fin function: asymptotic behavior of the filtered derivative for point processes in case of change points," Statistical Inference for Stochastic Processes, Springer, vol. 20(2), pages 253-272, July.
    2. Gut, Allan & Steinebach, Josef, 2010. "Asymptotics for increments of stopped renewal processes," Statistics & Probability Letters, Elsevier, vol. 80(7-8), pages 558-565, April.

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