Nonparametric modal regression

TL;DR

This paper introduces a simple nonparametric kernel density-based method for modal regression, which estimates the most probable outcomes of Y given X, providing insights beyond mean-based regression.

Contribution

It develops an asymptotic analysis, confidence and prediction set construction techniques, and discusses connections to related methods like mixture regression and density ridge estimation.

Findings

Derived asymptotic error bounds for the proposed method

Proposed techniques for confidence and prediction set construction

Connected modal regression to mixture models and density ridges

Abstract

Modal regression estimates the local modes of the distribution of $Y$ given $X=x$, instead of the mean, as in the usual regression sense, and can hence reveal important structure missed by usual regression methods. We study a simple nonparametric method for modal regression, based on a kernel density estimate (KDE) of the joint distribution of $Y$ and $X$. We derive asymptotic error bounds for this method, and propose techniques for constructing confidence sets and prediction sets. The latter is used to select the smoothing bandwidth of the underlying KDE. The idea behind modal regression is connected to many others, such as mixture regression and density ridge estimation, and we discuss these ties as well.