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Last edited on
Feb 06, 2021 by JJ
.
Thesis:
Master in Mathematics

Author:
Samuel Kilian

Title:
Minimax-Risiko bei der Schätzung von Untermannigfaltigkeiten unter Hausdorffabstand

Supervisor:
Jan JOHANNES

Abstract:
In this thesis we find lower and upper bounds for the minimax risk of estimating a manifold under two error models. The results are based on the article „Manifold Estimation and Singular Deconvolution under Hausdorff Loss“ by Genovese, Perone-Pacifico, Verdinelli and Wasserman (2012) and are partly deduced under altered conditions. The derivation of lower bounds require some terms of differential geometry, which are introduced, proved and illustrated. We will construct geometric objects explicitly and derive its properties in detail. Furthermore we show, that the partly modified estimators in Genovese, Perone-Pacifico, Verdinelli and Wasserman (2012) attain the lower bounds up to a logarithmic factor. Finally we see within the scope of a simulation study, that the estimator behaves regularly, when dealing with usual sample sizes.

Reference:
C. R. Genovese, M. Perone-Pacifico, I. Verdinelli and L. Wasserman. Manifold estimation and singular deconvolution under Hausdorff loss. The Annals of Statistics, 40(2):941–963, 2012.