A Tutorial on Kernel Density Estimation and Recent Advances

TL;DR

This tutorial reviews kernel density estimation (KDE), covering basic properties, confidence intervals, bias correction, and recent advances in geometric and topological inference, with practical R implementations included.

Contribution

It offers a comprehensive overview of KDE, integrating recent developments in confidence bands and topological inference, along with practical guidance and R code.

Findings

Convergence rates of KDE under various metrics

Methods for constructing confidence intervals and bands

Recent techniques for topological feature inference

Abstract

This tutorial provides a gentle introduction to kernel density estimation (KDE) and recent advances regarding confidence bands and geometric/topological features. We begin with a discussion of basic properties of KDE: the convergence rate under various metrics, density derivative estimation, and bandwidth selection. Then, we introduce common approaches to the construction of confidence intervals/bands, and we discuss how to handle bias. Next, we talk about recent advances in the inference of geometric and topological features of a density function using KDE. Finally, we illustrate how one can use KDE to estimate a cumulative distribution function and a receiver operating characteristic curve. We provide R implementations related to this tutorial at the end.