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.
Highlights
The problem of regression with functional predictors has been receiving increasing interests nowadays, boosted by more and more datasets with observations that can be naturally perceived as curves
This trend starts with the popular monograph Ramsay and Silverman (2002) that gives a detailed exposition of functional linear models
These two studies follow the parametric approach to functional regression, it is clear that nonparametric approach is a viable alternative (Lian, 2007)
Summary
The problem of regression with functional predictors has been receiving increasing interests nowadays, boosted by more and more datasets with observations that can be naturally perceived as curves. Nonparametric methods with functional predictors and scalar responses appear later (Ferraty and Vieu, 2002, 2004, 2006; Preda, 2007; Biau, Cerou and Guyader, 2010), which have been widely accepted by the statistical community as a more flexible approach to functional regression with fewer structural assumptions imposed As this area naturally develops and matures, the situation where the responses are curves begins to receive more. One might predict annual precipitation using temperature measurements as in Ramsay and Silverman (2005), or predict future hourly electricity consumption based on past history as in Antoch et al (2008) These two studies follow the parametric approach to functional regression, it is clear that nonparametric approach is a viable alternative (Lian, 2007). We note that throughout the paper we use C to denote a generic constant that assumes different values at different places
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