Penalized Spline Estimation for Generalized Partially Linear Single-Index Models

• Activity Number: 69
• Type: Contributed
• Date/Time: Sunday, August 7, 2005 : 4:00 PM to 5:50 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Yan Yu*+
• Companies: University of Cincinnati
• Keywords: Dimension Reduction ; Framingham Heart ; GLM ; GAM ; Inference ; Semiparametric

Abstract

Motivated by the famous Framingham Heart Study data, we propose penalized spline (P-spline) estimation for generalized partially linear single-index models where the systematic component has the form $\eta({\alpha}^T x) + {\beta}^T z$ and responses are from the general exponential family, such as binary or Poisson responses. Here, $\eta(\cdot)$ is an unknown univariate function estimated by P-splines. By reducing the dimensionality from that of a general covariate vector $x$ to a univariate index $\alpha^T x$, single-index models avoid the so-called ``curse of dimensionality.'' The P-spline approach offers a number of advantages. All parameters in the P-spline single-index model can be estimated simultaneously by penalized likelihood or quasi-likelihood. As a direct fitting method, our approach is rapid and computationally stable. Standard nonlinear optimization software can be used. Moreover, joint inference for $\eta(\cdot)$, $\alpha$, and $\beta$ is possible by standard estimating equations theory such as the sandwich formula for the joint covariance matrix.