##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability & Statistics

## Wanli Qiao

#### George Mason University

## Confidence Regions for Density Ridges

##### Abstract:

A ridge in a data cloud is a low-dimensional geometric feature that generalizes the concept of local modes, in the sense that ridge points are local maxima constrained in some subspace. In this talk we give nonparametric confidence regions for $r$-dimensional ridges of probability density functions on ${R}^d$ , 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. Two types of confidence regions for density ridges will be presented: one is based on the asymptotic distribution of the maximal vertical deviation, which is established by utilizing the extreme value theory of nonstationary $\chi^2$-fields indexed by manifolds; and the other is a bootstrap approach (including multiplier bootstrap and empirical bootstrap), the theoretical validity of which leverages the recent study in the literature on the Gaussian approximation of suprema of empirical processes.

Host: Ery Arias-Castro

### January 13, 2020

### 10:00 AM

### AP&M 6402

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