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Cramér‐von Mises tests for change points

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  • Rasmus Erlemann
  • Richard Lockhart
  • Rihan Yao

Abstract

We study two nonparametric tests of the hypothesis that a sequence of independent observations is identically distributed against the alternative that at a single change point the distribution changes. The tests are based on the Cramér–von Mises two‐sample test computed at every possible change point. One test uses the largest such test statistic over all possible change points; the other averages over all possible change points. Large sample theory for the average statistic is shown to provide useful p‐values much more quickly than bootstrapping, particularly in long sequences. Power is analyzed for contiguous alternatives. The average statistic is shown to have limiting power larger than its level for such alternative sequences. Evidence is presented that this is not true for the maximal statistic. Asymptotic methods and bootstrapping are used for constructing the test distribution. Performance of the tests is checked with a Monte Carlo power study for various alternative distributions.

Suggested Citation

  • Rasmus Erlemann & Richard Lockhart & Rihan Yao, 2022. "Cramér‐von Mises tests for change points," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(2), pages 802-830, June.
    • Handle: RePEc:bla:scjsta:v:49:y:2022:i:2:p:802-830
    DOI: 10.1111/sjos.12544
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    1. Duchesne, Pierre & Lafaye De Micheaux, Pierre, 2010. "Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods," Computational Statistics & Data Analysis, Elsevier, vol. 54(4), pages 858-862, April.
    2. McKeague, Ian W. & Sun, Yanqing, 1996. "Transformations of Gaussian random fields to Brownian sheet and nonparametric change-point tests," Statistics & Probability Letters, Elsevier, vol. 28(4), pages 311-319, August.
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