An Efficient Estimator for the Generalized Semilinear Model

Abstract

The generalized semilinear model (GSLIM) extends the generalized linear model by allowing the addition of an unknown smooth function to the predictor parameter, producing a semiparametric model. Some previously proposed score-based estimators for the parametric component in this model are not √ n consistent. We propose efficient estimators for both the parametric and nonparametric components of the GSLIM. The estimator for the parametric component is the solution to an estimated efficient score equation and is based on recent theoretical advances in semiparametric estimation. The estimator for the nonparametric component is the maximizer of an estimated penalized likelihood of the form considered by Cox and O'Sullivan. A robust standard error estimate maintains valid inference under certain model violations. An application and simulation results are given for data with binary outcomes, using a modified generalized cross-validation algorithm to select the smoothing parameter. Parameter and standard error estimates are within 6% of nominal values under a range of smooth functions tested in the simulations.