Abstract
We consider the problem of estimating the slope parameter in circular functional linear regression, where scalar responses Y 1, ..., Y n are modeled in dependence of 1-periodic, second order stationary random functions X 1, ...,X n . We consider an orthogonal series estimator of the slope function β, by replacing the first m theoretical coefficients of its development in the trigonometric basis by adequate estimators. We propose a model selection procedure for m in a set of admissible values, by defining a contrast function minimized by our estimator and a theoretical penalty function; this first step assumes the degree of ill-posedness to be known. Then we generalize the procedure to a random set of admissible m’s and a random penalty function. The resulting estimator is completely data driven and reaches automatically what is known to be the optimal minimax rate of convergence, in terms of a general weighted L 2-risk. This means that we provide adaptive estimators of both β and its derivatives.
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Comte, F., Johannes, J. Adaptive estimation in circular functional linear models. Math. Meth. Stat. 19, 42–63 (2010). https://doi.org/10.3103/S1066530710010035
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DOI: https://doi.org/10.3103/S1066530710010035
Key words
- orthogonal series estimation
- model selection
- derivatives estimation
- mean squared error of prediction
- minimax theory