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On the power of the Kolmogorov test to detect the trend of a Brownian bridge with applications to a change-point problem in regression models

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  • Bischoff, Wolfgang
  • Hashorva, Enkelejd
  • Hüsler, Jürg
  • Miller, Frank
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    Abstract

    Given a Brownian bridge B0 with trend g:[0,1]-->[0,[infinity]), Y(z)=g(z)+B0(z),z[set membership, variant][0,1],we are interested in testing H0:g[reverse not equivalent]0 against the alternative K:g>0. For this test problem we study weighted Kolmogorov testswhere c>0 is a suitable constant and w:[0,1]-->[0,[infinity]) is a weight function. To do such an investigation a recent result of the authors on a boundary crossing probability of the Brownian bridge is useful. In case the trend is large enough we show an optimality property for weighted Kolmogorov tests. Furthermore, an additional property for weighted Kolmogorov tests is shown which is useful to find the more favourable weight for specific test problems. Finally, we transfer our results to the change-point problem whether a regression function is or is not constant during a certain period.

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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 66 (2004)
    Issue (Month): 2 (January)
    Pages: 105-115

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    Handle: RePEc:eee:stapro:v:66:y:2004:i:2:p:105-115

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    Related research

    Keywords: Brownian bridge with trend Tests of Kolmogorov type Regression models Change-point problem;

    References

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    1. Jandhyala, V. K. & MacNeill, I. B., 1989. "Residual partial sum limit process for regression models with applications to detecting parameter changes at unknown times," Stochastic Processes and their Applications, Elsevier, vol. 33(2), pages 309-323, December.
    2. Wolfgang Bischoff & Frank Miller, 2000. "Asymptotically Optimal Tests and Optimal Designs for Testing the Mean in Regression Models with Applications to Change-Point Problems," Annals of the Institute of Statistical Mathematics, Springer, vol. 52(4), pages 658-679, December.
    3. Hackl, P & Westlund, A H, 1989. "Statistical Analysis of "Structural Change": An Annotated Bibliography," Empirical Economics, Springer, vol. 14(2), pages 167-92.
    4. Andrews, Donald W K, 1993. "Tests for Parameter Instability and Structural Change with Unknown Change Point," Econometrica, Econometric Society, vol. 61(4), pages 821-56, July.
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    Cited by:
    1. Bischoff, Wolfgang & Hashorva, Enkelejd, 2005. "A lower bound for boundary crossing probabilities of Brownian bridge/motion with trend," Statistics & Probability Letters, Elsevier, vol. 74(3), pages 265-271, October.

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