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.

本文提出了一种称为AFSSEN的新方法，该方法可以同时选择重要的预测变量并在具有次高斯误差的高维标量函数线性模型中生成平滑估计。假设结果位于一般的实际可分离希尔伯特空间中，$\mathbb{H}$，而参数位于一个称为Cameron-Martin空间的子空间中， $\mathbb{ķ}$，它与“复制内核希尔伯特空间”密切相关，因此参数估计会继承特定的属性，例如平滑度或周期性，而无需在数据上强制执行此类属性。我们提出了一种自适应弹性网罚分形式的正则化方法，该方法涉及混合两种类型的功能规范，从而为估计模型中的平滑和变量选择提供了微调控制。渐进理论以功能预言性的形式提供，并且本文以仿真研究作为结束，论证了在预测误差和变量选择方面，使用AFSSEN优于现有方法的优势。