Nonparametric modal regression in the presence of measurement error

TL;DR

This paper introduces two nonparametric methods for estimating the local modes of the conditional distribution of a response variable given a predictor with measurement error, analyzing their properties and comparing their performance.

Contribution

It proposes novel nonparametric estimators for conditional modes in the presence of measurement error, including implementation details and asymptotic analysis.

Findings

The proposed methods effectively estimate conditional modes with measurement error.

Numerical studies show improved performance over naive error-free methods.

Asymptotic properties of the estimators are established.

Abstract

In the context of regressing a response $Y$ on a predictor $X$, we consider estimating the local modes of the distribution of $Y$ given $X=x$ when $X$ is prone to measurement error. We propose two nonparametric estimation methods, with one based on estimating the joint density of $(X, Y)$ in the presence of measurement error, and the other built upon estimating the conditional density of $Y$ given $X=x$ using error-prone data. We study the asymptotic properties of each proposed mode estimator, and provide implementation details including the mean-shift algorithm for mode seeking and bandwidth selection. Numerical studies are presented to compare the proposed methods with an existing mode estimation method developed for error-free data naively applied to error-prone data.