  In this paper, we develop uniform inference methods for the conditional mode
based on quantile regression. Specifically, we propose to estimate the
conditional mode by minimizing the derivative of the estimated conditional
quantile function defined by smoothing the linear quantile regression
estimator, and develop two bootstrap methods, a novel pivotal bootstrap and the
nonparametric bootstrap, for our conditional mode estimator. Building on
high-dimensional Gaussian approximation techniques, we establish the validity
of simultaneous confidence rectangles constructed from the two bootstrap
methods for the conditional mode. We also extend the preceding analysis to the
case where the dimension of the covariate vector is increasing with the sample
size. Finally, we conduct simulation experiments and a real data analysis using
U.S. wage data to demonstrate the finite sample performance of our inference
method.
