Uniform in bandwidth rate of convergence of the conditional mode estimate on functional stationary ergodic data

The aim of this paper is to establish the uniform consistency with rate over a bandwidth interval of the kernel conditional mode estimate whenever functional stationary ergodic data are considered. This kind of result is immediately applicable to proving uniform consistency of kernel-type estimators when the bandwidth hh is a function of the data or the location xx. Notice that our uniform in bandwidth results are the first ones to be established in this setting. Moreover, the ergodic setting offers a more general framework in regards to the practice than the usual mixing structure.