Asymptotic Confidence Regions for Density Ridges

TL;DR

This paper introduces a large sample theory for constructing nonparametric confidence regions for density ridges in multivariate data, using extreme value theory of nonstationary chi-fields.

Contribution

It develops the first asymptotic confidence regions for density ridges based on the maximal deviation of kernel estimators constrained on the ridges.

Findings

Confidence regions are valid asymptotically.

Method applies to ridges of any dimension less than the ambient space.

Utilizes extreme value distribution of nonstationary chi-fields.

Abstract

We develop large sample theory including nonparametric confidence regions for $r$-dimensional ridges of probability density functions on $\mathbb{R}^d$, where $1\leq r<d$. We view ridges as the intersections of level sets of some special functions. The vertical variation of the plug-in kernel estimators for these functions constrained on the ridges is used as the measure of maximal deviation for ridge estimation. Our confidence regions for the ridges are determined by the asymptotic distribution of this maximal deviation, which is established by utilizing the extreme value distribution of nonstationary $\chi$-fields indexed by manifolds.