Dual-semiparametric regression using weighted Dirichlet process mixture

An efficient and flexible Bayesian approach is proposed for a dual-semiparametric regression model that models mean function semiparametrically and estimates the distribution of the error term nonparametrically. Using a weighted Dirichlet process mixture (WDPM), a Bayesian approach has been developed on the assumption that the distributions of the response variables are unknown. The WDPM approach is especially useful for real applications that have heterogeneous error distributions or come from a mixture of distributions. In the mean function, the unknown functions are estimated using natural cubic smoothing splines. For the error terms, several different WDPMs are proposed using different weights that depend on the distances between the covariates. Their marginal likelihoods are derived, and the computation of marginal likelihood for WDPM is provided. Efficient Markov chain Monte Carlo (MCMC) algorithms are also provided. The Bayesian approaches based on different WDPMs are compared with the parametric error model and the Dirichlet process mixture (DPM) error model in terms of the Bayes factor using a simulation study, suggesting better performance of the Bayesian approach based on WDPM. The advantage of the proposed Bayesian approach is also demonstrated using the credit rating data.