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October 2015 Asymptotic theory for density ridges
Yen-Chi Chen, Christopher R. Genovese, Larry Wasserman
Ann. Statist. 43(5): 1896-1928 (October 2015). DOI: 10.1214/15-AOS1329


The large sample theory of estimators for density modes is well understood. In this paper we consider density ridges, which are a higher-dimensional extension of modes. Modes correspond to zero-dimensional, local high-density regions in point clouds. Density ridges correspond to $s$-dimensional, local high-density regions in point clouds. We establish three main results. First we show that under appropriate regularity conditions, the local variation of the estimated ridge can be approximated by an empirical process. Second, we show that the distribution of the estimated ridge converges to a Gaussian process. Third, we establish that the bootstrap leads to valid confidence sets for density ridges.


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Yen-Chi Chen. Christopher R. Genovese. Larry Wasserman. "Asymptotic theory for density ridges." Ann. Statist. 43 (5) 1896 - 1928, October 2015.


Received: 1 June 2014; Revised: 1 March 2015; Published: October 2015
First available in Project Euclid: 3 August 2015

zbMATH: 1327.62303
MathSciNet: MR3375871
Digital Object Identifier: 10.1214/15-AOS1329

Primary: 62G20
Secondary: 62G05 , 62G15

Keywords: bootstrap , Density estimation , empirical process , nonparametric statistics , ridge

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 5 • October 2015
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