Rate of uniform consistency for a class of mode regression on functional stationary ergodic data

Abstract

The aim of this paper is to study the asymptotic properties of a class of kernel conditional mode estimates whenever functional stationary ergodic data are considered. To be more precise on the matter, in the ergodic data setting, we consider a random elements (X, Z) taking values in some semi-metric abstract space $E\times F$. For a real function $\varphi$ defined on the space F and $x\in E$, we consider the conditional mode of the real random variable $\varphi (Z)$ given the event “$X=x$”. While estimating the conditional mode function, say $\theta _\varphi (x)$, using the well-known kernel estimator, we establish the strong consistency with rate of this estimate uniformly over Vapnik–Chervonenkis classes of functions $\varphi$. Notice that the ergodic setting offers a more general framework than the usual mixing structure. Two applications to energy data are provided to illustrate some examples of the proposed approach in time series forecasting framework. The first one consists in forecasting the daily peak of electricity demand in France (measured in Giga-Watt). Whereas the second one deals with the short-term forecasting of the electrical energy (measured in Giga-Watt per Hour) that may be consumed over some time intervals that cover the peak demand.