Asymptotic theory for density ridges

TL;DR

This paper develops an asymptotic theory for density ridges, extending the understanding of density mode estimators to higher-dimensional structures, and provides methods for inference and confidence set construction.

Contribution

It introduces a comprehensive asymptotic framework for density ridges, including Gaussian approximation and bootstrap validity for inference.

Findings

Estimated ridge variation approximates an empirical process

Distribution of estimated ridge converges to a Gaussian process

Bootstrap provides valid confidence sets for 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.