This article considers the problem of nonparametric estimation of the regression function \(r\) in a functional regression model \(Y = r(X) +\varepsilon \) with a scalar response Y, a functional explanatory variable X, and a second order stationary error process \(\varepsilon \). Under some specific criteria, we construct a local linear kernel estimator of \(r\) from functional random design with correlated errors. The exact rates of convergence of mean squared error of the constructed estimator are established for both short and long range dependent error processes. Simulation studies are conducted on the performance of the proposed simple local linear estimator. Examples of time series data are considered.

本文考虑函数回归模型\(Y = r(X) +\varepsilon \)中回归函数 \(r\)的非参数估计问题，该模型具有标量响应Y、函数解释变量X和第二个变量阶平稳误差过程\(\varepsilon \)。在某些特定标准下，我们从具有相关误差的函数随机设计构建了\(r\)的局部线性核估计器。为短程和长程相关误差过程建立了所构造估计器的均方误差的准确收敛率。对所提出的简单局部线性估计器的性能进行了仿真研究。考虑时间序列数据的示例。