## Adaptive function-on-scalar regression with a smoothing elastic net.(English)Zbl 1470.62190

Summary: This paper presents a new methodology, called AFSSEN, to simultaneously select significant predictors and produce smooth estimates in a high-dimensional function-on-scalar linear model with sub-Gaussian errors. Outcomes are assumed to lie in a general real separable Hilbert space, $$\mathbb{H}$$, while parameters lie in a subspace known as a Cameron-Martin space, $$\mathbb{K}$$, which are closely related to Reproducing Kernel Hilbert Spaces, so that the parameter estimates inherit particular properties, such as smoothness or periodicity, without enforcing such properties on the data. We propose a regularization method in the style of an adaptive Elastic Net penalty that involves mixing two types of functional norms, providing a fine tune control of both the smoothing and variable selection in the estimated model. Asymptotic theory is provided in the form of a functional oracle property, and the paper concludes with a simulation study demonstrating the advantages of using AFSSEN over existing methods in terms of prediction error and variable selection.

### MSC:

 62R10 Functional data analysis 62H12 Estimation in multivariate analysis 62G08 Nonparametric regression and quantile regression 46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)

### Software:

flm; fda (R); Rcpp; R; NLopt
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