Bayesian analysis of generalized partially linear single-index models

We extend generalized partially linear single-index models by incorporating a random residual effect into the nonlinear predictor so that the new models can accommodate data with overdispersion. Based on the free-knot spline techniques, we develop a fully Bayesian method to analyze the proposed models. To make the models spatially adaptive, we further treat the number and positions of spline knots as random variables. As random residual effects are introduced, many of the completely conditional posteriors become standard distributions, which greatly facilitates sampling. We illustrate the proposed models and estimation method with a simulation study and an analysis of a recreational trip data set.