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Likelihood-based confidence bands for fault lines in response surfaces
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  • Published: September 2002

Likelihood-based confidence bands for fault lines in response surfaces

  • Peter Hall1 &
  • Christian Rau1 

Probability Theory and Related Fields volume 124, pages 26–49 (2002)Cite this article

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  • 6 Citations

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Abstract.

 We consider the problem of constructing asymptotic confidence bands, both pointwise and simultaneous, for a smooth fault line in a response surface when the design is represented by a point process, either deterministic or stochastic, with intensity n diverging to infinity. The estimator of the fault line is defined as the ridge line on the likelihood surface which arises from locally fitting a model that employs a linear approximation to the fault line, to a kernel smooth of the data. The construction is based on analysis of the limiting behaviour of perpendicular distance from a point on the true fault line to the nearest point on the ridge. We derive asymptotic properties of bias, and the limiting distribution of stochastic error. This distribution is given by the location of the maximum of a Gaussian process with quadratic drift. Although the majority of attention is focused on the regression problem, the limiting distribution is shown to have wider relevance to local-likelihood approaches to fault line estimation for density or intensity surfaces.

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Authors and Affiliations

  1. Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia. This work was supported by a grant on the APAC National Facility under the Merit Allocation Scheme. e-mail: Peter.Hall@anu.edu.au; Christian.Rau@anu.edu.au, , , , , , AU

    Peter Hall & Christian Rau

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  1. Peter Hall
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  2. Christian Rau
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Received: 20 August 2000 / Revised version: 27 September 2001 / Published online: 22 August 2002

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Hall, P., Rau, C. Likelihood-based confidence bands for fault lines in response surfaces. Probab Theory Relat Fields 124, 26–49 (2002). https://doi.org/10.1007/s004400100195

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  • Issue Date: September 2002

  • DOI: https://doi.org/10.1007/s004400100195

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Keywords

  • Response Surface
  • Point Process
  • Gaussian Process
  • Asymptotic Property
  • Fault Line
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