Asymptotic theory for density ridges
Yen-Chi Chen, Christopher R. Genovese, Larry Wasserman
The Annals of Statistics · 2015 · Volume 43 · Issue 5
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.
Keywords
62G05, 62G15, 62G20, bootstrap, Density estimation, empirical process, nonparametric statistics, ridge
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