Functional linear and single-index models are core regression methods in functional data analysis and are widely used for performing regression in a wide range of applications when the covariates are random functions coupled with scalar responses. In the existing literature, however, the construction of associated estimators and the study of their theoretical properties is invariably carried out on a case-by-case basis for specific models under consideration. In this work, assuming the predictors are Gaussian processes, we provide a unified methodological and theoretical framework for estimating the index in functional linear, and its direction in single-index models. In the latter case, the proposed approach does not require the specification of the link function. In terms of methodology, we show that the reproducing kernel Hilbert space (RKHS) based functional linear least-squares estimator, when viewed through the lens of an infinite-dimensional Gaussian Stein's identity, also provides an estimator of the index of the single-index model. Theoretically, we characterize the convergence rates of the proposed estimators for both linear and single-index models. Our analysis has several key advantages: (i) it does not require restrictive commutativity assumptions for the covariance operator of the random covariates and the integral operator associated with the reproducing kernel; and (ii) the true index parameter can lie outside of the chosen RKHS, thereby allowing for index misspecification as well as for quantifying the degree of such index misspecification. Several existing results emerge as special cases of our analysis.

翻译：函数线性模型与单指数模型是函数型数据分析中的核心回归方法，广泛应用于协变量为随机函数且响应为标量时的各类回归场景。然而现有文献中，这些模型的估计量构建及理论性质研究总是针对特定模型逐一进行。本文在预测变量服从高斯过程的假设下，为函数线性模型中的指标估计及单指数模型中的方向估计提供了统一的方法论与理论框架。对于单指数模型，所提方法无需指定链接函数。在方法论层面，我们证明了基于再生核希尔伯特空间（RKHS）的函数线性最小二乘估计量，通过无穷维高斯Stein恒等式的视角，同时提供了单指数模型指标的估计量。在理论层面，我们刻画了线性模型与单指数模型下所提估计量的收敛速率。本文分析具有以下关键优势：（i）无需对随机协变量的协方差算子与再生核相关的积分算子施加严格的交换性假设；（ii）真实指标参数可位于所选RKHS之外，从而允许指标设定偏差并量化该偏差程度。现有若干结果均为本文分析的特例。