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U-Statistics for Change under Alternatives

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  • Gombay, Edit

Abstract

Asymptotic distributions of U-statistics to test for possible changes in the distribution will be derived when the change occurred. We will show that for all possible types of kernels, symmetric, antisymmetric, degenerate, non-degenerate, the test statistics are asymptotically normally distributed. We also study the distribution of the estimator of the time of change. Its large sample behaviour is approximately that of the maximum of a two-sided random walk. The terms in these random walks explain the exact nature of bias in the change-point estimator. Several examples will be given as illustration.

Suggested Citation

  • Gombay, Edit, 2001. "U-Statistics for Change under Alternatives," Journal of Multivariate Analysis, Elsevier, vol. 78(1), pages 139-158, July.
    • Handle: RePEc:eee:jmvana:v:78:y:2001:i:1:p:139-158
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    References listed on IDEAS

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    1. Hall, Peter, 1979. "On the invariance principle for U-statistics," Stochastic Processes and their Applications, Elsevier, vol. 9(2), pages 163-174, November.
    2. Ferger, Dietmar & Stute, Winfried, 1992. "Convergence of changepoint estimators," Stochastic Processes and their Applications, Elsevier, vol. 42(2), pages 345-351, September.
    3. Dietmar Ferger, 1994. "On the power of nonparametric changepoint-tests," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 41(1), pages 277-292, December.
    4. Csörgo, Miklós & Horváth, Lajos, 1988. "Invariance principles for changepoint problems," Journal of Multivariate Analysis, Elsevier, vol. 27(1), pages 151-168, October.
    5. Ferger, D., 1994. "An Extension of the Csörgo-Horváth Functional Limit Theorem and Its Applications to Changepoint Problems," Journal of Multivariate Analysis, Elsevier, vol. 51(2), pages 338-351, November.
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    Cited by:

    1. Alfredas Račkauskas & Martin Wendler, 2020. "Convergence of U-processes in Hölder spaces with application to robust detection of a changed segment," Statistical Papers, Springer, vol. 61(4), pages 1409-1435, August.
    2. Dominique Guegan, 2007. "Global and local stationary modelling in finance: theory and empirical evidence," Post-Print halshs-00187875, HAL.
    3. Lajos Horváth & Gregory Rice, 2014. "Extensions of some classical methods in change point analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 219-255, June.

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