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.

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