In this work, we consider the estimation of the parametric component of a functional single-index model that relates the functional covariate, which is a sample-path of a zero mean Gaussian process, to a scalar-valued response. Based on infinite-dimensional extension of Gaussian Stein's identity, which allows to estimate the parametric component while being oblivious to the non-parametric component, we propose an estimator that involves solving a ridge regression problem in a reproducing kernel Hilbert space (RKHS). We establish convergence rates for this estimator in the regimes of commutativity and non-commutativity of the kernel integral operator and the covariance operator of the Gaussian process. These results also recover the optimal rates for prediction error when the model is linear and the operators commute.
Joint work with Krishnakumar Balasubramanian (UC Davis) and Hans-Georg Mueller (UC Davis)