Statistica Sinica 14(2004), 449-455

ROOT-N CONSISTENCY OF PENALIZED SPLINE

ESTIMATOR FOR PARTIALLY LINEAR SINGLE-INDEX

MODELS UNDER GENERAL EUCLIDEAN SPACE

Yan Yu and David Ruppert

University of Cincinnati and Cornell University

Abstract: 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 $\eta_{0}(\alpha_{0}^T{\bf x}) + {\beta_0}^T \hbox{\bf z}.$ This approach is computationally stable and efficient in practice. Furthermore, it yields a root-n consistent estimate of the single-index parameter $\alpha$ 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.

Key words and phrases: Asymptotics, compact, inference, nonparametric, ridge regression.