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MEAN SQUARE ERROR IN REGRESSION ESTIMATION WITH FUNCTIONAL DATA
PAULO EDUARDO OLIVEIRA
Dedicated to Nazar´e Mendes Lopes on the occasion of her sixtieth birthday
Abstract. The performance of kernel estimators is mainly a↵ected by the choice of the bandwidth parameter. Characterizations for the band- width are usually based on convenient descriptions of the mean square error. So, we prove a characterization of this error criterium, for regres- sion estimation in a functional framework, assuming only the continuity of the regression function. The representation depends on the behaviour of small-ball probabilities, and highlights the influence of the geometry and smoothness of the distribution of the functional variable in the de- scription of this mean square error.
1. Introduction
The problem of approximating regression functions is one of the most classi- cal in statistics and data analysis. Regression estimation is most usually studied in Rd, where the Lebesgue measure plays an essential role, due to its invari- ance to translations. We will be looking at regression estimation considering a functional framework, thus placing ourselves in an infinite dimensional space where no analogue of the Lebesgue measure exists. This functional framework has received a lot of interest from statisticians in recent years. First steps into this direction were made by Ge↵roy [7, 8] who studied the regressogram, later followed by a point process approach by Jacob and Oliveira [10, 11], although not really exploring the functional framework. A really functional ap- proach was used later by Ramsay and Silverman [17] for some case studies,
Accepted: 10 April 2015.
2010 Mathematics Subject Classification. 62G05, 62G08, 62M09.
Key words and phrases. functional data, regression estimation, small-ball probability, mean square error.
This work was partially supported by the Centre for Mathematics of the Univer- sity of Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Part- nership Agreement PT2020.
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