Asymptotic normality of the Nadaraya–Watson estimator for nonstationary functional data and applications to telecommunications

Abstract

We study a nonparametric regression model, where the explanatory variable is nonstationary dependent functional data and the response variable is scalar. Assuming that the explanatory variable is a nonstationary mixture of stationary processes and general conditions of dependence of the observations (implied in particular by weak dependence), we obtain the asymptotic normality of the Nadaraya-Watson estimator. Under some additional regularity assumptions on the regression function, we obtain asymptotic confidence intervals for the regression function. We apply this result to estimate the quality of service for an end-to-end connection on a network.

Asymptotic normality of a robust estimator of the regression function for functional time series data

“…and strong mixing). Among the recent lot of papers on the modelization of variables taking values in infinite dimensional spaces, we refer to the papers by Aspirot, Bertin, and Perera (2009), Delsol (2009), Ferraty, Rabhi, and Vieu (2005, Laïb and Louani (submitted for publication) among others.…”

Nonparametric kernel regression estimation for functional stationary ergodic data: Asymptotic properties

“…Asymptotic issues for functional data have recently received an increasing interest, one may refer to [20,13,3,10,11,22,23,21,19,9,2,8,7] and to the recent monograph by Ferraty and Vieu [12] and the references therein.…”

Dimension reduction in functional regression using mixed data canonical correlation analysis

“…In this article, we focus on functional regression with a scalar response and a functional predictor. The basic approach to analyzing such data is functional linear regression [11,17,22,28], while more flexible functional regression models can be found in [2,5,16,27] and many others. A thorough discussion on this issue is given in [24,Chapter 15], and more comprehensive reviews of functional regression models can be found in Ramsay and Silverman [23,24] and Ferraty and Vieu [10].…”

“…Therefore, we consider regression-based dimension reduction and use a set of inner products to represent the reduced dimensions. More specifically, we focus on the following model (2) y = f ( β 1 , x , . .…”

“…A number of methods have been proposed for finding β i 's in (2) following the idea of sliced inverse regression (SIR) [21] for multivariate predictors, including functional SIR (FSIR) [7], functional inverse regression (FIR) [8], wavelet smoothing (WS) [1], and regularized FSIR (RFSIR) [9]. The aforementioned methods are all able to consistently estimate the EDR directions in that they guarantee that the estimated directions are contained in the EDR space.…”