Asymptotic theory for density ridges

Abstract

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.

Published at http://dx.doi.org/10.1214/15-AOS1329 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (6)

• Gaussian approximation of suprema of empirical processes
• The topography of multivariate normal mixtures
• Anti-concentration and honest, adaptive confidence bands
• Nonparametric ridge estimation
• Statistical Analysis of Metric Graph Reconstruction
• Theoretical Analysis of Nonparametric Filament Estimation

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