Abstract: We propose a semiparametric functional single-index model for studying the relationship between a univariate response and multiple functional covariates. The parametric part of the model integrates a functional linear regression model and a sufficient dimension-reduction structure. The nonparametric part of the model allows the response-index dependence or the link function to be unspecified. The B-spline method is used to approximate the coefficient function, which leads to a dimension-folding-type model. A new kernel regression method is developed to handle the dimension-folding model, allowing us to estimate the index vector and the B-spline coefficients efficiently. We also establish the asymptotic properties and semiparametric optimality for the estimators.

Key words and phrases: B-spline, dimension folding, dimension reduction, functional data analysis, functional linear model, kernel estimation.