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

Abstract

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 h is a function of the data or the location x. 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.