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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


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


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.

Suggested Citation

  • Bischoff, Wolfgang & Hashorva, Enkelejd & Hüsler, Jürg & Miller, Frank, 2004. "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," Statistics & Probability Letters, Elsevier, vol. 66(2), pages 105-115, January.
  • Handle: RePEc:eee:stapro:v:66:y:2004:i:2:p:105-115

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    References listed on IDEAS

    1. 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;The Institute of Statistical Mathematics, vol. 52(4), pages 658-679, December.
    2. 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.
    3. Andrews, Donald W K, 1993. "Tests for Parameter Instability and Structural Change with Unknown Change Point," Econometrica, Econometric Society, vol. 61(4), pages 821-856, July.
    4. Hackl, P & Westlund, A H, 1989. "Statistical Analysis of "Structural Change": An Annotated Bibliography," Empirical Economics, Springer, vol. 14(2), pages 167-192.
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    Cited by:

    1. Deng, Pingjin, 2017. "Boundary non-crossing probabilities for Slepian process," Statistics & Probability Letters, Elsevier, vol. 122(C), pages 28-35.
    2. 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.
    3. Pingjin Deng, 2016. "Asymptotic of Non-Crossings probability of Additive Wiener Fields," Papers 1610.07131,


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