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 eta(0)(alpha(0) (T) x) + beta(0) (T) z. This approach is computationally stable and efficient in practice. Furthermore, it yields a root-n consistent estimate of the single-index parameter or and the partially linear parameter beta 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.