  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.
