Functional regression allows for a scalar response to be dependent on a functional predictor, however, not much work has been done when scalar predictors that interacts with the functional predictor are introduced. In this paper, we introduce a new functional single-index varying coefficient model with the functional predictor being single-index part. By means of functional principal components analysis and basis function approximation, we obtain the estimators of slope function and coefficient functions, and propose an iterative estimating procedure. Furthermore, the rates of convergence of the proposed estimators and the mean squared prediction error are established under some regularity conditions. At last, we illustrate the finite sample performance of our proposed methods with some simulation studies and a real data application.

功能回归允许标量响应依赖于功能预测器，但是，当引入与功能预测器交互的标量预测器时，没有做太多的工作。在本文中，我们介绍了一种新的功能性单指标变化系数模型，其中功能预测变量为单指标部分。通过功能主成分分析和基函数逼近，我们得到了斜率函数和系数函数的估计量，并提出了一个迭代估计程序。此外，在某些规则性条件下，确定了估计量的收敛速度和均方预测误差。最后，通过一些仿真研究和实际数据应用，说明了所提出方法的有限样本性能。