Root-n Consistency of Penalized Spline Estimator for Partially Linear Single-index Models under General Euclidean Space

Single-index models are important in multivariate nonparametric regression. In a previous paper, we proposed a penalized spline approach to a partially linear single-index model where the mean function has the form η0(α0 T x) +β0 T z. This approach is computationally stable and efficient in practice. Furthermore, it yields a root-n consistent estimate of the single-index parameter α and the partially linear parameter β with a nontrivial smoothing parameter under the assumption of a compact parameter space. In this paper, we relax the compactness assumption and prove the existence and root-n consistency of the constrained penalized least squares estimators. We expect our proof technique to be useful for establishing asymptotic properties of the penalized spline approach to other model fitting.