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Maximin Estimation of Multidimensional Boundaries

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  • Muller, H. G.
  • Song, K. S.

Abstract

We consider the problem of estimating the location and size of a discontinuity in an otherwise smooth multidimensional regression function. The boundary or location of the discontinuity is assumed to be a closed curve respective surface, and we aim to estimate this closed set. Our approach utilizes the uniform convergence of multivariate kernel estimators for directional limits. Differences of such limits converge to zero under smoothness assumptions, and to the jump size along the discontinuity. This leads to the proposal of a maximin estimator, which selects the boundary for which the minimal estimated directional difference among all points belonging to this boundary is maximized. It is shown that this estimated boundary is almost surely enclosed in a sequence of shrinking neighborhoods around the true boundary, and corresponding rates of convergence are obtained.

Suggested Citation

  • Muller, H. G. & Song, K. S., 1994. "Maximin Estimation of Multidimensional Boundaries," Journal of Multivariate Analysis, Elsevier, vol. 50(2), pages 265-281, August.
    • Handle: RePEc:eee:jmvana:v:50:y:1994:i:2:p:265-281
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    Cited by:

    1. Polzehl, Jörg & Spokojnyj, Vladimir G., 1998. "Image denoising: Pointwise adaptive approach," SFB 373 Discussion Papers 1998,38, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    2. Bengs, Viktor & Eulert, Matthias & Holzmann, Hajo, 2019. "Asymptotic confidence sets for the jump curve in bivariate regression problems," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 291-312.
    3. Cheng, Ming-Yen & Hall, Peter, 2006. "Methods for tracking support boundaries with corners," Journal of Multivariate Analysis, Elsevier, vol. 97(8), pages 1870-1893, September.
    4. Garlipp, T. & Muller, C.H., 2006. "Detection of linear and circular shapes in image analysis," Computational Statistics & Data Analysis, Elsevier, vol. 51(3), pages 1479-1490, December.
    5. Delgado, Miguel A. & Hidalgo, Javier, 2000. "Nonparametric inference on structural breaks," Journal of Econometrics, Elsevier, vol. 96(1), pages 113-144, May.
    6. Peihua Qiu, 2009. "Jump-preserving surface reconstruction from noisy data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(3), pages 715-751, September.

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