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 α0Tx with a nonparametric component, η0(α0Tx), where η0(·) is an unknown univariate link function. By reducing the dimensionality from that of a general covariate vector x to a univariate index α0Tx, single-index models avoid the so-called “curse of dimensionality.” We propose penalized spline (P-spline) estimation of η0(·) in partially linear single-index models, where the mean function has the form η0(α0Tx) + β 0Tz. 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 η0(·), α0, and β0 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 √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. GeneralLq penalty functions can be readily implemented.