Electronic Journal of Statistics

Convergence of nonparametric functional regression estimates with functional responses

Heng Lian

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Abstract

We consider nonparametric functional regression when both predictors and responses are functions. More specifically, we let $(X_{1},Y_{1}),\ldots, (X_{n},Y_{n})$ be random elements in $\mathcal{F}\times\mathcal{H}$ where $\mathcal{F}$ is a semi-metric space and $\mathcal{H}$ is a separable Hilbert space. Based on a recently introduced notion of weak dependence for functional data, we showed the almost sure convergence rates of both the Nadaraya-Watson estimator and the nearest neighbor estimator, in a unified manner. Several factors, including functional nature of the responses, the assumptions on the functional variables using the Orlicz norm and the desired generality on weakly dependent data, make the theoretical investigations more challenging and interesting.

Article information

Source
Electron. J. Statist., Volume 6 (2012), 1373-1391.

Dates
First available in Project Euclid: 26 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1343310301

Digital Object Identifier
doi:10.1214/12-EJS716

Mathematical Reviews number (MathSciNet)
MR2988451

Zentralblatt MATH identifier
1295.62042

Subjects
Primary: 62G08: Nonparametric regression 60G10: Stationary processes

Keywords
Bernstein’s inequality for martingale differences Nadaraya-Watson estimate nearest neighbor estimate nonparametric functional regression Orlicz norm

Citation

Lian, Heng. Convergence of nonparametric functional regression estimates with functional responses. Electron. J. Statist. 6 (2012), 1373--1391. doi:10.1214/12-EJS716. https://projecteuclid.org/euclid.ejs/1343310301


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