Bayesian estimation and variable selection for single index models

We develop a fully Bayesian method to analyze the single index models, including variable selection, the index vector estimation and the link function fitting with free-knot splines. The proposed method is implemented by means of the reversible jump Markov chain Monte Carlo technique. We treat the marginal posterior of all the unknown quantities except the spline coefficients and error variance as the target distribution to reduce the dimension of the parameters and to obtain a rapid algorithm. We design a new random walk Metropolis sampler to sample from the conditional posterior distribution of the index vector. The proposed method is verified by simulation studies, and is applied to analyze two real data sets.