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

Abstract

a b s t r a c tWe propose a family of robust nonparametric estimators for a regression function based on the kernel method. We establish the asymptotic normality of the estimator under the concentration property on small balls probability measure of the functional explanatory variable when the observations exhibit some kind of dependence. This approach can be applied in time series analysis to make prediction and build confidence bands. We illustrate our methodology on the US electricity consumption data.

Nonparametric M-Estimation for Functional Spatial Data

“…Therefore, we obtain the following result which is exactly the same as obtained by Attouch et al (2010) Corollary 3. Under assumptions (H1)∼(H7), we have:…”

“…The main result of this work is that, under general mixing assumptions, the estimator considered is asymptotically normally distributed. Notice that, this work extends to spatial case the results given by Attouch et al (2010) in functional time series data.…”

“…The robust analysis in functional time series, which is a particular case of functional random filed (N = 1), has been investigated by many authors. We cite, for example, Crambes et al (2008) for the convergence in L q norm, Attouch et al (2010) for the asymptotic normality and Chen and Zhang (2009) for the weak and strong consistency of the nonparametric functional conditional location estimate in the mixing case.…”

Robust Regression for Functional Time Series Data

“…case. We discuss how this asymptotic results is required to obtaining the results stated by Crambes et al (2008) or Attouch et al (2010). To illustrate the importance of this work in practice, we clarify how this asymptotic results can be applied to the functional time series prediction which provides natural prediction tool to outlier diagnostics.…”

“…and strong mixing). The asymptotic normality of this last model has been established by Attouch et al (2010), under the concentration properties on small balls of the probability measure of the underlying functional variable. Among the recent lot of papers concerning the modelization of variable taking values in infinite dimensional spaces, we only refer to the papers by Dabo-Niang andRhomari (2009), Ferraty et al (2010) and Ezzahrioui and Ould Saïd (2010).…”