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June 2016 Theoretical analysis of nonparametric filament estimation
Wanli Qiao, Wolfgang Polonik
Ann. Statist. 44(3): 1269-1297 (June 2016). DOI: 10.1214/15-AOS1405


This paper provides a rigorous study of the nonparametric estimation of filaments or ridge lines of a probability density $f$. Points on the filament are considered as local extrema of the density when traversing the support of $f$ along the integral curve driven by the vector field of second eigenvectors of the Hessian of $f$. We “parametrize” points on the filaments by such integral curves, and thus both the estimation of integral curves and of filaments will be considered via a plug-in method using kernel density estimation. We establish rates of convergence and asymptotic distribution results for the estimation of both the integral curves and the filaments. The main theoretical result establishes the asymptotic distribution of the uniform deviation of the estimated filament from its theoretical counterpart. This result utilizes the extreme value behavior of nonstationary Gaussian processes indexed by manifolds $M_{h},h\in(0,1]$ as $h\to0$.


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Wanli Qiao. Wolfgang Polonik. "Theoretical analysis of nonparametric filament estimation." Ann. Statist. 44 (3) 1269 - 1297, June 2016.


Received: 1 May 2014; Revised: 1 October 2015; Published: June 2016
First available in Project Euclid: 11 April 2016

zbMATH: 1338.62139
MathSciNet: MR3485960
Digital Object Identifier: 10.1214/15-AOS1405

Primary: 62G20
Secondary: 62G05

Keywords: extreme value distribution , integral curves , kernel density estimation , Nonparametric curve estimation

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 3 • June 2016
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