A new sparse semiparametric model is proposed, which incorporates the influence of two functional random variables in a scalar response in a flexible and interpretable manner. One of the functional covariates is included through a single-index structure, while the other is included linearly through the high-dimensional vector formed by its discretised observations. For this model, two new algorithms are presented for selecting relevant variables in the linear part and estimating the model. Both procedures utilise the functional origin of linear covariates. Finite sample experiments demonstrated the scope of application of both algorithms: the first method is a fast algorithm that provides a solution (without loss in predictive ability) for the significant computational time required by standard variable selection methods for estimating this model, and the second algorithm completes the set of relevant linear covariates provided by the first, thus improving its predictive efficiency. Some asymptotic results theoretically support both procedures. A real data application demonstrated the applicability of the presented methodology from a predictive perspective in terms of the interpretability of outputs and low computational cost.

翻译：提出了一种新的稀疏半参数模型，该模型以灵活且可解释的方式将两个函数型随机变量对标量响应的影响纳入考量。其中一个函数协变量通过单指标结构引入，而另一个则通过其离散观测值形成的高维向量以线性方式纳入。针对该模型，提出了两种用于在线性部分选择相关变量并估计模型的新算法。两种算法均利用了线性协变量的函数来源。有限样本实验展示了两种算法的应用范围：第一种方法是一种快速算法，能够（在不损失预测能力的情况下）解决标准变量选择方法在估计该模型时所需的大量计算时间问题；第二种算法则补全了第一种算法提供的相关线性协变量集，从而提升了其预测效率。一些渐近结果从理论上为两种算法提供了支撑。一项实际数据应用从输出可解释性和低计算成本的角度，展示了所提出方法在预测方面的适用性。