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.

On the Behavior of Extreme d-dimensional Spatial Quantiles Under Minimal Assumptions

“…Pick then n large enough to have α n ≥ η and µ α n ,u n ≥ R (existence follows from Theorem 2). By definition, this implies that O P α n ,u n ( µ α n ,u n w n ) = O P α n ,u n (µ α n ,u n ) ≤ O P α n ,u n ( µ α n ,u n u n ), which contradicts (7). Therefore, it is sufficient to prove (7).…”

“…The success of spatial quantiles is partly explained by their ability to cope with high-dimensional data and even functional data; see, e.g., [3], [4], [5] and [6]. These quantiles were also used with much success to conduct multiple-output quantile regression, again also in the framework of functional data analysis; we refer to [7], [9], and [10]. The present work, however, focuses on the finite-dimensional case.…”

Multiple‐output quantile regression through optimal quantization

“…These quantiles are suitable in large dimensions and even in general Hilbert spaces, which make them applicable to high‐dimensional or functional data. They have been used in a quantile regression framework, even in the functional case (e.g., Chakraborty, ; Cheng & De Gooijer, ; Chaouch & Laïb, , ; Chowdhury & Chaudhuri, ). Although they can be made affine‐equivariant through a transformation‐retransformation approach (see Chakraborty, , ), geometric (regression) quantiles intrinsically are orthogonal‐equivariant only.…”

Nonparametric regression method with functional covariates and multivariate response

“…For instance, Matsui et al (2008) illustrates the association between multiple scalar responses and functional predictors by using the Gaussian basis function. The problem with multivariate response variables and functional covariates based on the L 1 -median regression estimation is described by Chaouch and Laïb (2013). Wang and Chen (2015) proposes the formulation of the covariance function for the multi-response Gaussian process regression.…”