Randomly censored quantile regression estimation using functional stationary ergodic data

Abstract

This paper investigates the conditional quantile estimation of a randomly censored scalar response variable given a functional random covariate (i.e. valued in some infinite-dimensional space) whenever a stationary ergodic data are considered. A kernel-type estimator of the conditional quantile function is introduced. Then, a strong consistency rate as well as the asymptotic distribution of the estimator are established under mild assumptions. A simulation study is considered to show the performance of the proposed estimator. An application to the electricity peak demand prediction using censored smart meter data is also provided.

CLT for single functional index quantile regression under dependence structure

“…As one can see, the asymptotic variance Σ(θ, ζ θ (γ, x), x) depends on some unknown functions f(θ, ζ θ (γ, x), x) and φ θ,x (h K ) and other theoretical quantities F(θ, ζ θ (γ, x), x), G(•) and ζ θ (γ, x) that have to be estimated in practice. Therefore, G(•), F(θ, t, x) and ζ θ (γ, x) should be replaced, respectively, by the Kaplan-Meier's estimator G n (•), the kernel-type estimator of the joint distribution f(θ, ζ θ (γ, x), x) and ζ θ,n (γ, x) the conditional quantile estimator given by equation (8). Moreover, using the decomposition given by assumption (H1), one can estimate φ θ,x (z) by F x,n (z) = 1/n n i=1 1 {X i ∈B θ (x,z)} .…”

“…This section considers simulated as well as real data studies to assess the finitesample performance of the proposed estimator and compare it to its competitor. More precisely, we are interested in comparing the conditional quantile estimator based on single functional index model (SFIM) to the kernel-type conditional quantile estimator (NP) introduced in Chaouch and Khardani [8] when the data is dependent and the response variable is subject to a random right-censorship phenomena. Throughout the simulation part, the n i.i.d.…”

“…The training subsample is used to estimate the single functional index and to select the smoothing parameters h k and h H . Whereas the testing subsample is used to assess and compare the single functional index based estimator of the conditional quantile, namely ζ θ (γ, •), to the kernel-type conditional quantile estimator, say ζ(γ, •), which is introduced in Chaouch and Khardani [8] as follows:…”

“…Unlike the regression function (which is defined as the conditional mean) that relies only on the central tendency of the data, the conditional quantile function allows the analysts to estimate the functional dependence between variables for all portions of the conditional distribution of the response variable. Moreover, quantiles are well known for their robustness to heavy-tailed error distributions and outliers which allow to consider them as a useful alternative to the regression function Chaouch and Khardani [8].…”

Asymptotic Results of a Recursive Double Kernel Estimator of the Conditional Quantile for Functional Ergodic Data

“…On the other hand, the convergence in L p −norm stated by Dabo-Niang and Laksaci [8]. The interested reader can also refer to some of the following additional references [18], [22] and [7] to expand further on this topic and take an overview.…”