Penalized spline estimation for partially linear single-index models

Single-index models are potentially important tools for multivariate nonparametric regression. They generalize linear regression by replacing the linear combination alpha(0)(T)x with a nonparametric component, eta(0)(alpha(0)(T)x), where eta(0)(.) is an unknown univariate link function. By reducing the dimensionality from that of a general covariate vector x to a univariate index alpha(0)(T)x, single-index models avoid the so-called curse of dimensionality. We propose penalized spline (P-spline) estimation of eta(0)((.))in partially linear single-index models, where the mean function has the form eta(0)(alpha(0)(T)x) + beta(0)(T)z. The P-spline approach offers a number of advantages over other fitting methods for single-index models. All parameters in the P-spline,single-index model can be estimated simultaneously by penalized nonlinear least squares. As a direct least squares fitting method, our approach is rapid and computationally stable. Standard nonlinear least squares software can be used. Moreover, joint inference for eta(0)(.), alpha(0), and beta(0) is possible by standard estimating equations theory such as the sandwich formula for the joint covariance matrix. Using asymptotics where the number of knots is fixed, though potentially large, we show squareroot of n consistency and asymptotic normality of the estimators of all parameters. These asymptotic results permit joint inference for the parameters. Several examples illustrate that the model and proposed estimation methodology can be effective in practice. We investigate inference based on the sandwich estimate through a Monte Carlo study. General L-q penalty functions can be readily implemented.