# Nonparametric Mode Regression Estimation for Functional Stationary Ergodic Data. Asymptotic Normality and Application

Published in Mathematical Methods of Statistics, 2016

### S. Bouzebda, M. Chaouch and N. Laïb

The main purpose of the present work is to establish the functional asymptotic normality of a class of kernel conditional mode estimates whenever functional stationary ergodic data are considered. More precisely, consider a random variable $(X, Z)$ taking values in some semi-metric abstract space $E\times F$. For a real function $\varphi$ defined on $F$ and for each $x\in E$, we consider the conditional mode, say $\Theta_\varphi(x)$, of the real random variable $\varphi(Z)$ given the event $X=x"$. While estimating the conditional mode function by $\widehat{\Theta}_{\varphi, n}(x)$, using the kernel-type estimator, we establish the limiting law of the family of processes $\{\widehat{\Theta}_\varphi(x)-\Theta_\varphi(x)\}$ (suitably normaliezd) over Vapnik-Chervonenkis class ${\cal C}$ of functions $\varphi$. Beyond ergodicity, any other assumption is imposed on the data. This paper extends the scope of some previous results established under mixing condition for a fixed function $\varphi$. From this result, the asymptotic normality of a class of predictors is derived and confidence bands are constructed. Finally, a general notion of bootstrapped conditional mode constructed by exchangeably weighting samples is presented. The usefulness of this result will be illustrated in the construction of confidence bands.