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.
Haiming Zhou. Xianzheng Huang. "Nonparametric modal regression in the presence of measurement error." Electron. J. Statist. 10 (2) 3579 - 3620, 2016. https://doi.org/10.1214/16-EJS1210