Nonparametric multivariate $L_{1}$-median regression estimation with functional covariates

Abstract

In this paper, a nonparametric estimator is proposed for estimating the $L_{1}$-median for multivariate conditional distribution when the covariates take values in an infinite dimensional space. The multivariate case is more appropriate to predict the components of a vector of random variables simultaneously rather than predicting each of them separately. While estimating the conditional $L_{1}$-median function using the well-known Nadarya-Waston estimator, we establish the strong consistency of this estimator as well as the asymptotic normality. We also present some simulations and provide how to built conditional confidence ellipsoids for the multivariate $L_{1}$-median regression in practice. Some numerical study in chemiometrical real data are carried out to compare the multivariate $L_{1}$-median regression with the vector of marginal median regression when the covariate $X$ is a curve as well as $X$ is a random vector.