# 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.

2016-03-01-Bouzebda-Chaouch-Laib.md