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Asymptotics of a Boundary Crossing Probability of a Brownian Bridge with General Trend

Author

Listed:
  • Wolfgang Bischoff

    (University of Karlsruhe)

  • Frank Miller

    (University of Karlsruhe)

  • Enkelejd Hashorva

    (University of Bern)

  • Jürg Hüsler

    (University of Bern)

Abstract

Let us consider a signal-plus-noise model γh(z)+B 0(z), z ∈ [0,1], where γ > 0, h: [0,1] → ℝ, and B 0 is a Brownian bridge. We establish the asymptotics for the boundary crossing probability of the weighted signal-plus-noise model for γ→∞, that is P (sup zε [0,1] w(z)(γ h(z)+B 0(z))>c), for γ→∞, (1) where w: [0,1]→ [0,∞ is a weight function and c>0 is arbitrary. By the large deviation principle one gets a result with a constant which is the solution of a minimizing problem. In this paper we show an asymptotic expansion that is stronger than large deviation. As byproduct of our result we obtain the solution of the minimizing problem occurring in the large deviation expression. It is worth mentioning that the probability considered in (1) appears as power of the weighted Kolmogorov test applied to the test problem H 0: h≡ 0 against the alternative K: h>0 in the signal-plus-noise model.

Suggested Citation

  • Wolfgang Bischoff & Frank Miller & Enkelejd Hashorva & Jürg Hüsler, 2003. "Asymptotics of a Boundary Crossing Probability of a Brownian Bridge with General Trend," Methodology and Computing in Applied Probability, Springer, vol. 5(3), pages 271-287, September.
  • Handle: RePEc:spr:metcap:v:5:y:2003:i:3:d:10.1023_a:1026242019110
    DOI: 10.1023/A:1026242019110
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    Citations

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    Cited by:

    1. Hashorva, Enkelejd & Jaworski, Piotr, 2012. "Gaussian approximation of conditional elliptical copulas," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 397-407.
    2. Deng, Pingjin, 2017. "Boundary non-crossing probabilities for Slepian process," Statistics & Probability Letters, Elsevier, vol. 122(C), pages 28-35.
    3. Hashorva, Enkelejd, 2019. "Approximation of some multivariate risk measures for Gaussian risks," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 330-340.
    4. Pingjin Deng, 2016. "The joint distributions of running maximum of a Slepian processes," Papers 1609.04529, arXiv.org.
    5. Pingjin Deng, 2016. "Asymptotic of Non-Crossings probability of Additive Wiener Fields," Papers 1610.07131, arXiv.org.
    6. Pingjin Deng, 2016. "The boundary non-Crossing probabilities for Slepian process," Papers 1608.01133, arXiv.org.
    7. Pingjin Deng, 2018. "The Joint Distribution of Running Maximum of a Slepian Process," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1123-1135, December.
    8. Enkelejd Hashorva, 2010. "Boundary Non-crossings of Brownian Pillow," Journal of Theoretical Probability, Springer, vol. 23(1), pages 193-208, March.
    9. 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.
    10. Enkelejd Hashorva & Yuliya Mishura & Georgiy Shevchenko, 2021. "Boundary Non-crossing Probabilities of Gaussian Processes: Sharp Bounds and Asymptotics," Journal of Theoretical Probability, Springer, vol. 34(2), pages 728-754, June.
    11. E. Hashorva, 2018. "Approximation of Some Multivariate Risk Measures for Gaussian Risks," Papers 1803.06922, arXiv.org, revised Oct 2018.

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